Lesson 1 Introduction

Outline Statistics Descriptive versus inferential statistics Population versus Sample Statistic versus Parameter Simple Notation Summation Notation Statistics What are statistics? What do you thing of when you think of statistics? Can you think of some examples where you have seen statistics used? You might think about where in the real world you see statistics being used, or think about how statistics in used in your major. Statistics are divided into two main areas: descriptive and inferential statistics. Descriptive statistics- These are numbers that are used to consolidate a large amount of information. Any average, for example, is a descriptive statistic. So, batting averages, average daily rainfall, or average daily temperature are good examples of descriptive statistics. Inferential statistics- inferential statistics are used when we want to draw conclusions. For example when we want to determine if some treatment is better than another, or if there are differences in how two groups perform. A good book definition is using samples to draw inferences about populations. More on this once we define samples and populations. Population- Any set of people or objects with something in common. Anything could be a population. We could have a population of college students. We might be interested in the population of the elderly. Other examples include: single parent families, people with depression, or burn victims. For anything we might be interested in studying we could define a population. Very often we would like to test something about a population. For example, we might want to test whether a new drug might be effective for a specific group. It is impossible most of the time to give everyone a new treatment to determine if it worked or not. Instead we commonly give it to a group of people from the population to see if it is effective. This subset of the population is called a sample. When we measure something in a population it is called a parameter. When we measure something in a sample it is called a statistic. For example, if I got the average age of parents in single-family homes, the measure would be called a parameter. If I measured

the age of a sample of these same individuals it would be called a statistic. Thus, a population is to a parameter as a sample is to a statistic. This distinction between samples and population is important because this course is about inferential statistics. With inferential statistics we want to draw inferences about populations from samples. Thus, this course is mainly concerned with the rules or logic of how a relatively small sample from a large population could be tested, and the results of those tests can be inferred to be true for everyone in the population. For example, if we want to test whether Bayer asprin is better than Tylonol at relieving pain, we could not give these drugs to everyone in the population. Its not practical since the general population is so large. Instead we might give it to a couple of hundred people and see which one works better with them. With inferential statistics we can infer that what was true for a few hundred people is also true for a very large population of hundreds of thousands of people. When we write symbols about populations and samples they differ too. With populations we will use Greek letters to symbolize parameters. When we symbolize a measure from a sample (a statistic) we will use the letters you are familiar with (Roman letters). Thus, if I measure the average age of a population Id indicate the value with the Greek letter mu ( =24). While if I were to measure the same value for a subset of the population or a sample then I would indicate the value with a roman letter ( X =24). Simple Notation You might thing about descriptive statistics as the vocabulary of the "language" of statistics. If this is true then summation notation can be thought of as the alphabet of that language. Notation and summation notation is just a short hand way of representing information we have collected and mathematical operation we want to perform. For example, if I collect data on a variable, say the amount of time (in minutes) several people spent waiting at a bus stop, I can represent that group of numbers with the variable X. The variable X represents all of the data that I collected.

Amount of Time

X 5.0 11.1 8.9 3.5 12.3 15.6

With subscripts I can also represent an individual data point within the variable set we have labeled X. For example the third data point, 8.9, is the X3 data point. The fifth data point X5 is the number 12.3. Very often when we want to represent ALL of the data

points in a variable set we will use X by itself, but we may also add the subscript i. Whenever you the subscript i, you can assume that we are referring to all the numbers for the variable X. Thus, Xi is all of the numbers in the data set or: 5,11.1,8.9,3.5,12.3,15.6. There are other common symbols we will use besides X. Sometimes we will have two data sets to deal with and refer to one distribution as X and the other distribution as Y. It is also necessary for many formulas to know how many data points are in a data set. The symbol for the number of data points in a set is N. For the data set above the number of data points or N = 6. In addition, we will use the average or mean value a good deal. We will indicate the mean, as noted above, differently for the population () than for the sample ( X ). Summation Notation Another common symbol we will use is the summation sign ( ). This symbol does not represent anything about our data itself, but instead is an operation we must perform. Whenever you see this symbol it means to add up whatever appears to the right of the sign. Thus, X or Xi tells us to add up all of the data points in our data set. For our example above it would be: 5 + 11.1 + 8.9 + 3.5 + 12.3 + 15.6 = 56.4. You will see the summation sign with other mathematical operations as well. For example X2 tells us to add all the squared X values. Thus, for our example: X2 = 52 + 11.12 + 8.92 + 3.52 + 12.32 + 15.62 -or- 25 + 123.21 + 79.21 + 12.25 + 151.29 + 243.36 = 634.32. A few more examples of summation notation are in order since the summation sign will be central to the formulas we write. The following examples should give you a better idea about how the summation sign is used. Be sure you recall the order of operations needed to solve mathematical expressions. You will find a review on the web page or you can click here: http://faculty.uncfsu.edu/dwallace/sorder.html For the examples below we will use a new distribution. X = 1 2 3 4 Y = 5 6 7 8

( )22 XX

For this expression we are saying that the sum of the squared Xs is not equal to the sum of the Xs squared. Notice here we want to perform the operation in parentheses first, and then the exponents, and then the addition. Thus:

( )22 XX ( )22222 43214321 ++++++

1 + 4 + 9 + 16 (10)2 30 100 For the next expression we show, like in algebra, that the law of distribution applies to the summation sign as well. Again, what is important is to get a feel for how the summation sign works in equations.

YXYX +=+ )( (1+5)+(2+6)+(3+7)+(4+8) = (1+2+3+4)+(5+6+7+8) 6 + 8 + 10 + 12 = 10 + 26 36 = 36

Lesson 2 Scales of Measure

Outline Variables -measurement versus categorical -continuous versus discreet -independent and dependent Scales of measure -nominal, ordinal, interval, ratio Variables A variable is anything we measure. This is a broad definition that includes most everything we will be interested in for an experiment. It could be the age or gender of participants, their reactions times, or anything we might be interested in. Whenever we measure a variable, it could be a measurement (quantitative) difference or a categorical (qualitative) difference. You should know both terms for each type. Measurement variables are things to which we can assign a number. It is something we can measure. Examples include age, height, weight, time measurement, or number of children in a household. These examples are also called quantitative because they measure some quantity. Categorical variables are measures of differences in type rather than amount. Examples include anything categorize such as race, gender, or color. These are also called qualitative variables because there is some quality that distinguishes these objects. Another dimension on which variables might differ is that they may be either continuous or discreet. A continuous variable is a variable that can take on any value on the scale used to measure it. Thus, a measure of 1 or 2 is valid, as well as 1.5 or 1.25. Any division on any unit on the scale produces a valid possible measure. Examples include things like height or weight. You could have an object that weighed 1 pound or 1.5 pounds or 1.25 pounds. All are possible measures. Discreet variables, on the other hand, can assume only a few possible values on the scale used to measure it. Divisions of measures are usually not valid. Thus, if I measure the number of television sets in your home it could be 1 or 2 or 3. Divisions of these values are not valid. So, you could not have 1.5 televisions or 1.25 televisions in your home. You either have a television or you dont. Another way to keep this difference in mind is that with a continuous variable is a measure of how much. A discreet variable is a measure of how many. Scales of Measure whenever we measure a variable it has to be on some type of scale. The following scales are delivered in order of increasing complexity. Each scale presented is in order of increasing order. Nominal scales These are not really scales as all, but are instead numbers used to differentiate objects. Real world examples of these variables are common. The numbers

are just labels. So, social security numbers, the channels on your television, and sports team jerseys are all good examples of nominal variables. Ordinal Scales Ordinal scales use numbers to put objects in order. No other information other than more or less is available from the scale. A good example is class rank, or any type of ranking. Someone ranked at four had a higher GPA than someone ranked as five, but we dont know how much better four is than five. Interval Scales- Interval scales contain an ordinal scale (objects are in order), but have the added feature that the distance between scale units is always the same. Class rank would not qualify because we dont know how much better one unit is than another, but with interval there is the same distance from one unit to the next anywhere we are on the scale. Examples include temperature (in Fahrenheit or Celsius), or altitude. For temperature you know that the difference in ten degrees is the same no matter how hot or cold it might be. Ratio Scales Ratio scales contain an interval scale (equal intervals between units on the scale), but have the added feature that there is a true zero point on the scale. This zero point is necessary for ratio statements to have meaning. Examples include height or weight or measures of amount of time. Notice that it is not valid to have a measure below zero on any of these scales. Something could not weigh a negative amount. These scales are much more common than interval scales because if a scale usually has a zero point. In fact scientist invented the Kelvin temperature scale so that they would have a measure of temperature on a ratio scale. Again, in order to make ratio statements such as something is twice or half of another then it must be a variable on a ratio scale.

Lesson 3 Data Displays

Outline Frequency Distributions Grouped Frequency Distributions -class interval and frequency -cumulative frequency -relative percent -cumulative relative percent -interpretations Histograms/Bar Graphs Frequency Distributions We often form frequency distributions as a way to abbreviate the values we are dealing with in a distribution. With frequency distributions we will simply record the frequency or how many values fall at a particular point on the scale. For example, if I record the number of trips out of town (X) a sample of FSU students makes, I might end up with the following data: 0 2 5 3 2 4 3 1 0 2 6 0 4 7 0 1 2 4 3 5 4 3 1 6 1 0 5 3 Instead of having a jumbled set of numbers, we can record how many of each value (f) there are for the entire x-distribution. Below is a simple frequency distribution where the X column represents the number of trips, and the corresponding value for f indicates how many people in the sample gave us that particular response. X f 0 5 1 4 2 4 3 5 4 4 5 3 6 2 7 1 From the graph we can see that five people took no trips out of town, four people took one trip out of town, four people took two trips out of town, and so on. It is important not to confuse the f-value and the x-value. The f-values are just a count of how many. So, you can reverse the process as well. It might also be helpful in some examples to go from a frequency distribution back to original data set, especially if it causes confusion.

In the following example I start with a frequency distribution and go backward to find all the original values in the distribution. X f 0 2 1 3 2 4 3 3 4 2 What is the most frequent score? The answer is two because we will have four twos in our distribution: 0 0 1 1 1 2 2 2 2 3 3 3 4 4 Grouped Frequency Distributions The above examples used discreet measures, but when we measure a variable it is often on a continuous scale. In turn, there will be few values we measure that are at the exact same point on the scale. In order to build the frequency distribution we will group several values on the scale together and count any of measurements we observe in that range for the frequency. For example, if we measure the running time of rats in a maze we might obtain the following data. Notice that if I tried to count how many values fall at any single point on the scale my frequencies will all be one. 3.25 3.95 4.61 5.92 6.87 7.12 7.58 8.25 8.69 9.56 9.67 10.24 10.95 10.99 11.34 11.59 12.34 13.45 14.53 14.86 We will begin by forming the class interval. This will be the range of value on the scale we include for each interval. There are many rules we could use to determine the size of the interval, but for this course I will always indicate how big the interval should be. In the end, we want to construct a display that has between 5 and 15 intervals. Thus: Class Interval 0-2 3-5 6-8 9-11 12-14 Once we have the class interval, we will count how many values fall within the range of each interval. Since there is a gab in each class interval, we will be actually counting any values that would get rounded down or up into a particular interval. For example, with

the above data the value 8.26 would be rounded down into the 6-8 class interval. The value 8.69 would be rounded up into the 9-11 class interval. We will include a column to indicate the real limits of the class interval. These are the limits of the interval, including any rounded values. Real Limits Class Interval f -.5-2.5 0-2 0 2.5-5.5 3-5 3 5.5-8.5 6-8 5 8.5-11.5 9-11 7 11.5-14.5 12-14 5 Notice that my real limits cover half the distance of the gap between each class interval. Most of the time this value will be 0.5 since most scales will have one unit values and 0.5 is half the distance. So, real limits have no gap, but the class intervals do. If a value falls exactly on one of the real limits we could randomly choose its group. Cumulative Frequency Once we have formed the basic grouped frequency distribution above, we can add more columns for more detailed information. The first of these is the cumulative frequency column. With this column we will keep a running count of the frequency column as we move down the class interval. Real Limits Class Interval f Cum. f -.5-2.5 0-2 0 0 2.5-5.5 3-5 3 3 5.5-8.5 6-8 5 8 8.5-11.5 9-11 7 15 11.5-14.5 12-14 5 20 So, at the first interval we have zero frequency, so cumulatively we have zero values. For the second interval we have three, so cumulatively we have three. For the third interval we have five values, so cumulatively we have 8. That includes the five for the third interval, plus the three from the previous intervals. We continue this process until the last interval. Notice that when we reach the last interval we have all the values in the distribution represented. So, the bottom cumulative frequency is N or the total number of values in the distribution (20 here).

Relative Percent Another column will tell us the proportion of total values that fall at each interval. That is, we will express the frequency (column) as a percentage of the total. To convert the frequency to a percentage take the frequency (f) and divide by the number of values (N). This will give us the proportion of values for that particular interval. Move the decimal over two places (or multiply by 100) to change the proportion into a percent. Thus: Real Limits Class Interval f Cum. f Rel % -.5-2.5 0-2 0 0 0 2.5-5.5 3-5 3 3 15 5.5-8.5 6-8 5 8 25 8.5-11.5 9-11 7 15 35 11.5-14.5 12-14 5 20 25 Cumulative Relative Percent For a final column we will keep a running count of the relative percent column in the same way we did with the cumulative frequency. Keep in mind we are counting relative percentages now as we move down the display. Real Limits Class Interval f Cum. f Rel % Cum. Rel. % -.5-2.5 0-2 0 0 0 0 2.5-5.5 3-5 3 3 15 15 5.5-8.5 6-8 5 8 25 40 8.5-11.5 9-11 7 15 35 75 11.5-14.5 12-14 5 20 25 100 Notice that we can keep a running count of the relative percent column, but we could also obtain the same numbers by computing the percentage for each cumulative frequency as well. Interpretations The data display gives a good deal of information about where values in the sample fall. One good piece of information is about percentiles. A percentile is the percentage at or below a certain score. You often get percentile information when you get your SAT or ACT test scores back. Percentile information is found in the cumulative relative percentage column. Each value in that column tells us the percentage of the distribution at that point or less on the scale. Since we will be rounding values down into a certain interval based on the real limits, then we will indicate where on the scale a certain percentile is based on its corresponding upper real limit. For example, what score corresponds with the 75th percentile? The answer is 11.5 because any values of 11.5 or less are within the bottom 75% of the distribution. Similarly, what percentile is associated with a score of 8.5? We would use the cumulative relative percent that corresponds to 8.5, which is 40%. So, the score 8.5 corresponds with the bottom 40% or 40th percentile of the distribution. Other interpretations from the table can be made as well. For example, we might be interested in how many people fall at a particular interval, or at or below a certain

interval. How many scored between 3 and 5? The answer is a found in the frequency column, or three. How many scored 8.5 or less? The answer for this question is in the cumulative frequency column, or eight. Histograms/Bar Graphs We can also take the frequency information in our frequency or grouped frequency distribution and form a graph. In the graph we will form a simple x-y axis. On the x-axis we will place values from our scale, and on the y-axis we will plot the frequency for each point on the scale. For grouped frequency distributions, we will use the midpoint of each interval to indicate different points on the scale. We will continue with our previous example, but notice I have created a new column that indicates the center or midpoint of each interval. We will use this value to graph the display. Real Limits Class Interval MP f Cum. f Rel % Cum. Rel. % -.5-2.5 0-2 1 0 0 0 0 2.5-5.5 3-5 4 3 3 15 15 5.5-8.5 6-8 7 5 8 25 40 8.5-11.5 9-11 10 7 15 35 75 11.5-14.5 12-14 13 5 20 25 100

Note that the bars are touching. The bars touch like this when we are dealing with continuous data rather than discreet data. When the scale measures discreet values we call it a bar graph, and the lines do not touch. For example, if I measured the number of democrats, republicans, and independents in a sample, we would use a bar graph if we wanted to create a data display.

0100

200

300

400

500

600

Dem Rep Ind

Party

Frequency

Lesson 4 Measures of Central Tendency

Outline Measures of a distributions shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Measures of Shape With frequency distribution you can an idea of a distributions shape. If we trace the outline of the edges of the frequency bars you can idea about the shape.

From this point on, I will draw these shapes to illustrate different point throughout the semester. Keep in mind what you are looking at is a line indicating the frequency or how many values in a distribution lie at a particular point on the scale: just like a histogram.

Modality measures the number of major peaks in a distribution. A single major peak is unimodal, and is the most common modality. Two major peaks is a bi-modal distribution. You could also have multi-modal distributions.

Skewness measures the symmetry of a distribution. A symmetric distribution is most common, and is not skewed. If the distribution is not symmetric, and one side does not reflect the other, then it is skewed. Skewness is indicated by the tail or trailing frequencies of the distribution. If the tail is to the right it is a positive skew. If the tail is to the left then it is a negatively skewed distribution. For example, a positively skewed distribution would be: 1, 1, 2, 2, 2, 3, 3, 3, 9, 10. The outliers are on the high end of the scale. On the other hand a negatively skewed distribution might be: 1, 2, 9, 9, 9, 10, 10, 10, 11, 11, 11. Here the outliers are on the low end of the scale.

The normal distribution is one that is unimodal and symmetric. Most things we can measure in nature exhibit a normal distribution of values. Regression toward the mean is an idea that states values will tend to cluster around the mean with few values toward the trailing ends or tails of the distribution. As a result, most things we measure will tend to have a normal shape. Think about measures of height. There are very few people that are extremely tall or extremely short, but most tend to cluster around the average. With I.Q. scores, measures of weight, or most anything we can measure, the same pattern will repeat. Since most things we measure have more values close to the mean, we end up with mostly normally shaped distributions.

Measures of Central Tendency Knowing where the center of a distribution is tells us a lot about a distribution. Thats because most of the scores in a distribution will tend to cluster about the center. Measures of central tendency give us a number that describes where the center lies (and most scores as well). Mean The mean, or average score, is the arithmetic center of the distribution. You can find the mean by adding all the scores ( X ) together and dividing by the number of values you added together (N). X 1 2 3 4 5 15=X N = 5

For the Population: = XN

= 5

15 = 3

For the Sample: x = Xn

= 5

15 = 3

Note that we calculate the mean the same way for both the sample and the population we symbolize them differently. Other statistics will differ in how they are computed for the sample versus the population. Most students are familiar with these measures of central tendency, but there are several properties that may be new to you. 1) The first property of the mean is that it is the most reliable and most often used measure of central tendency. 2) The second property of the mean is that it need not be an actual score in the distribution. 3) The third property is that it is strongly influenced by outliers. 4) The fourth property is that the sum of the deviations about the mean must always equal zero. The last two properties need further explanation. An outlier is an extreme score. It is a score that lies apart from most of the rest of the distribution. If there are several outliers in a distribution it will often result in skewed shape to the distribution. Outliers tend to pull central tendency measures with them. Thus a distribution of values 1, 2, 3, 4, 5 has an average of 3. Three does a good job of describing where most of the scores in this distribution lie. However, if there is an outlier, say by substituting 25 for the 5 in the above distribution, then the mean changes a great deal. The new distribution 1, 2, 3, 4, 25 has a mean of 7. Seven is not really close to most of the other values in the distribution. Thus, the mean is a poor measure of the center when we have outliers, or a skewed distribution. A deviation is just a difference. A deviation from the mean is the difference between a score and the mean. So, when we say the sum of the deviations about the mean must always equal zero is just a way of saying that there are just as many differences between values above the mean and the center as there are differences between values below the mean and the center. Thus, for a simple distribution 1, 2, 3, 4, 5 the average is 3. Lets use population symbols and say = 3. The deviations are the differences between the score and the mean. X X- 1 1-3 = -2 2 2-3 = -1 3 3-3 = 0 4 4-3 = 1 5 5-3 = 2 = 3 )( X = 0 Now if we add these deviations we will always get zero, no matter what original values we use. This concept will be important when we consider standard deviation because we will need to look at differences between values in our distribution and the mean. Median The median is the physical center of the distribution. It is the value in the middle when the values of the distribution are arranged sequentially. The distribution:

1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7 has a median value of 4 because there are five values above this point and five values below this point in the distribution (1, 2, 2, 3, 4, 5, 5, 5, 6, 7). If you have an even set of numbers then there will be two values that are at the center, and you average these two values together in order to determine the median. For example, if we take out one of the numbers in the distribution so that we have 1, 1, 2, 2, 3, 4, 5, 5, 5, 6 then the two values in the center are 3 and 4 (1, 1, 2, 2, 3, 4, 5, 5, 5, 6). The average is 3.5 and that is the median. The median is resistant to outliers. That is, outliers will generally not affect the median and it will not be affected as much as the mean. It is possible the median might move slightly in the direction of the skew or outliers in the distribution. Mode The mode is the most frequent value in the distribution. It is simply the value that appears most often. For example in the distribution: 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 6, 7 there is only one mode (4). But, in the distribution: 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 6 there are two modes (3 and 4). If there is only one of each value then there is no mode. The mode is not affected by outliers. Since it is only determined by one point on the scale, other values will have no effect. Skewness and Central Tendency We have already discussed how each measure is affected by outliers or skewed distribution. Lets consider this information further. In a positively skewed distribution the outliers will be pulling the mean down the scale a great deal. The median might be slightly lower due to the outlier, but the mode will be unaffected. Thus, with a negatively skewed distribution the mean is numerically lower than the median or mode.

The opposite is true for positively skewed distributions. Here the outliers are on the high end of the scale and will pull the mean in that direction a great deal. The median might be slightly affected as well, but not the mode. Thus, with a positively skewed distribution the mean is numerically higher than the median or the mode.

Lesson 5 Measures of Dispersion

Outline Measures of Dispersion - Range - Interquartile Range - Standard Deviation and Variance Measures of Dispersion Measure of central tendency give us good information about the scores in our distribution. However, we can have very different shapes to our distribution, yet have the same central tendency. Measures of dispersion or variability will give us information about the spread of the scores in our distribution. Are the scores clustered close together over a small portion of the scale, or are the scores spread out over a large segment of the scale? Range The range is the difference between the high and low score in a distribution. Simply subtract the two numbers to find the range. So, in the distribution: 1, 3, 5, 9, 11 the range is 11 1 = 10. Remember to subtract the two numbers to give one number for the final answer. Interquartile Range (IQR) The interquartile range is the range of the middle 50% of a distribution. Because any outliers in our distribution must be on the ends of the distribution, the range as a measure of dispersion can be strongly influenced by outliers. One solution to this problem is to eliminate the ends of the distribution and measure the range of scores in the middle. Thus, with the interquartile range we will eliminate the bottom 25% and top 25% of the distribution, and then measure the distance between the extremes of the middle 50% of the distribution that remains. To actually compute some IQRs we would need to use calculus. Instead, of that possibility we will use a method that will yield a consistent, and somewhat accurate answer. Before we compute the value, lets learn some new definitions. A quartile is a quarter or 25% of the distribution. When we compute the IQR we will want to find each of the quartiles. The first quartile is the same as the 25th percentile because 25 percent of the distribution is at or below that point. The second quartile is the same thing as the 50th percentile and the median. The third quartile is the same as the 75th percentile. The IQR is the found by eliminating the values that lie between the bottom end and the first quartile (bottom 25%). We will also eliminate the values between the third quartile and the top of the distribution. We then subtract the new low and high score of the left over middle part of the distributions. So, IQR = Quartile 3 Quartile 1 or IQR = 75th percentile 25th percentile. To compute the IQR first arrange your numbers from lowest to highest and 1) find the median. The median is the 50th percentile and second quartile. Its a starting point for us to find the other quartiles. 2) Next find the median of the bottom half of the distribution

(ignoring the top half). This value is the 25th percentile or first quartile because we have taken the bottom 50% and cut it in half. 3) Find the median of the top half of the distribution just like we did for the bottom. This value is the 75th percentile or third quartile. 4) Next subtract the upper and lower medians you found in step 2 and 3. In the following example the median is 8 because it is the average of the two middle numbers. The value 8 is the 50th percentile or second quartile, though we will not use this number in the computation.

1 2 5 6 7 9 10 12 15 19 Once we find the median we can divide the distribution into two halves

1 2 5 6 7 9 10 12 15 19 Bottom 50% Top 50% The median of the bottom 50% is 5. The median of the top 50% is 12. So, IQR = 12 5 = 7

Bottom 25% Top 25%

1 2 5 6 7 9 10 12 15 19

Standard Deviation and Variance While the interquartile range eliminates the problem of outliers it creates another problem in that you are eliminating half of your data. Generally, this is not acceptable because of the difficulty in collecting data in the first place. The solution to both problems is to measure variability from the center of the distribution. Both standard deviation and variance measure how far on average scores deviate or differ from the mean. In this way, we use all the values in our data to compute variability, and outliers will not have undue influence.

Middle 50%

To compute standard deviation and variance we first start by finding the deviation about the mean. Recall that we did the same thing when discussing properties of the mean. Ill use the same example with the simple distribution 1, 2, 3, 4, 5. First we find the mean and the deviations about the mean. What we want to do is add up these deviations and find out how far on average the scores deviate from the mean. The problem we run into is that whenever we add the deviations (in order to find the average of the deviations) they will always sum to zero. How can we get an average if the sum is always zero? X X- 1 1-3 = -2 2 2-3 = -1 3 3-3 = 0 4 4-3 = 1 5 5-3 = 2 = 3 )( X = 0 One solution is to square all of the deviations. When we square all the numbers the negative values will all become positive and we can then add the deviations without getting zero. X X- (X )2 1 1-3 = -2 4 2 2-3 = -1 1 3 3-3 = 0 0 4 4-3 = 1 1 5 5-3 = 2 4 = 3 )( X = 0 2)( X = 10 Once we add the squared deviations we have a measure of overall variability in the distribution. The sum of the average squared deviations is called the sums of squares, and will be used in almost everyone formula we learn this semester. Please refer back to this section if formulas give you problems later on in the course. Once we divide these squared sums we will get the average squared deviation or variance. In this example it is 10/5 = 2. Since we are in squared units and not the same units as our scale we can take the square root of the variance in order to get the standard deviation. The standard deviation is the average deviation about the mean. For our example we take the square root of 2 and find 1.41 is the standard deviation.

The formula that contains all these operations is as follows. Note that 2 is just the symbol we use for population variance and is the symbol we use to denote population standard deviation.

( )22N

= X

= 2 population variance population standard deviation When dealing with a sample a minor change to the formula is made, and instead of subtracting the numerator by N, we divide by n 1. Try the numbers in the above example to compute the sample variance and standard deviation (variance is 2.5, standard deviation is 1.58).

( )221=

nXX

S

s = s2 sample variance sample standard deviation Please review the animated demonstration on variance and standard deviation for another example of how the population formula works. In addition an alternative formula for these same computations is presented. Although the formula detailed here is the best for understanding the concept, the one presented on the web page will be easier to use in the long run. Both appear in the homework packet formula section as well. See http://faculty.uncfsu.edu/dwallace/ssandrd1.html

Lesson 6 Z-Scores

Outline Linear Transformation -effect of addition/subtraction -effect of multiplication/division Z-Transformation Z-Distribution -properties Using Z-scores to compare values Linear Transformation Anytime we change a distribution by using a constant we perform a linear transformation. For example if I measure the heights of everyone registered in this course, but then found the tape measure I was using was missing the first two inches (so it started at inch two instead of zero), what would I have to do to find the true heights of everyone? If you think about it you will see that I must subtract two inches from each measurement to get the true heights (because the start position was too high). This example of a linear transformation is one in which we simply shift the numbers up on the same scale. Notice that even though all the numbers move, the relationship between values is not affected. X X + 2 55 57 57 59 58 510 510 60 You will need to know how linear transformations affect the mean and standard deviation of a distribution as well. How does adding or subtracting a constant affect the mean and standard deviation? How does multiplying and dividing a constant affect the mean and standard deviation?

When adding or subtracting a constant from a distribution, the mean will change by the same amount as the constant. The standard deviation will remain unchanged. This fact is true because, again, we are just shifting the distribution up or down the scale. We do not affect the distance between values. In the following example, we add a constant and see the changes to the mean and standard deviation. X X +5 1 6 2 7 3 8 4 9 5 10 = 3 = 8 = 1.41 = 1.41 The effect is a little different when we multiply or divide by a constant. For these transformations the mean will change by the same amount as the constant, but this time the standard deviation will change too. That is because when we multiply numbers together, for example, we change the distance between values rather than just shifting them up or down the scale. In the following example, we multiply a constant and see the changes to the mean and standard deviation. X X * 5 1 5 2 10 3 15 4 20 5 25 = 3 = 15 = 1.41 = 7.91 Z-Transformation The z-transformation is a linear transformation, just like those we have discussed. Transforming a raw score to a Z-score will yield a number that expresses exactly how many deviations from the mean a score lays. Here is the formula for transforming a raw

score in a population to a Z-score: z = X

Notice that the distance a score lies from the mean is now relative to how much scores deviate in general from the mean in the population. Regardless of what the raw score values are in the population, when we use the Z-transformation we obtain a standard measure of the distance of the score from the mean. Anytime Z=1, the raw score that produced the Z is exactly one standard deviation from the mean for that population. Anytime Z=1.5, the raw score that produced the Z is exactly 1.5 standard deviations from the mean for that population.

Think for a minute about what it means to know how many standard deviations from a mean a score lays. Consider our simple distribution example. X 1 2 3 4 5 = 3 = 1.41 What z-score will we expect the value 3 to have in this example? That is, how many standard deviations from the mean is 3? The answer is that it is at the mean, so it is zero standard deviations from the mean and we will get a z-score of zero for the original value of three. Now consider the value 1 in the distribution. What z-score will we expect to get for this score? Will it be less than one standard deviation or more than one standard deviation away from the mean? You can estimate the z-score by counting from the mean. One standard deviation is 1.41 units. Counting down from the mean the value 2 is one unit from the mean. Thats a little less than one standard deviation. We have to go down 1.41 units from the mean before we reach one standard deviation. So, when we get to 1 on the scale, we are two units from the mean and a little more than one standard deviation below the mean. Lets transform the simple distribution into a distribution of z-scores by plugging each value into the z-formula: X Z-Tranformation Z

1 ==41.1

31z -1.42

2 z = X = -.71

3 z = X = 0

4 z = X = .71

5 z = X = 1.42

= 3 = 0 = 1.41 = 1

The value of 1 is 1.42 standard deviations below the mean. The value 2 is .71 standard deviations below the mean, and so on. Notice that the mean is at zero, so any scores below the mean in the original distribution will always have a negative z-score and any score above the mean will have a positive z-score. Properties of the z-distribution. Also notice in the above example that we had to compute the mean and standard deviation of the simple x-distribution in order to compute the z-score. We can compute the mean and standard deviation of the resulting z-distribution as well. The mean of the z-distribution will always be zero, and the standard deviation will always be one. These facts make sense because the mean is always zero standard deviations away from the mean, and units on the z-distribution are themselves standard deviations. Using Z-scores to Compare Values Since z-scores reflect how far a score is from the mean they are a good way to standardize scores. We can take any distribution and express all the values as z-scores (distances from the mean). So, no matter scale we originally used to measure the variable, it will be expressed in a standard form. This standard form can be used to convert different scales to the same scale so that direct comparison of values from the two different distributions can be directly compared. For example, say I measure stress in the average college sophomore on a scale between 0 and 30 and find the mean is 15. Another researcher measures stress with the same population, but uses a scale that ranges from 0 to 300 with a mean of 150. Who has more stress, a person with a stress score of 15 from the first distribution or a person with a stress score of 150 from the second group? The value of 150 is a much larger number than 15, but both are at the mean of their own distribution. Thus, they will both have the same z-score of zero. Both values indicate an average amount of stress.

Consider another example. Lets say that Joan got an x = 88 in a class that had a mean score of 72 with a standard deviation of 10 ( = 72, = 10).

In a different class lets say Bob got a x = 92. The mean for Bobs class, however, was 87 with a standard deviation of 5 ( = 87, = 5). Who had the higher grade relative to their class? If you think about it for a second you will know that Joans score of 88 is much higher relative to the average of 72 compared to Bobs score of 92 to the average of 87. We can easily compare the values, however, if we simply compute the z-score for each. Joan

6.1

1016

107288 ===Z

Joan has the higher score because she is 1.6 standard deviations above the mean, and Bobs score is only 1 standard deviation above the mean. Bob

1

55

58792 ===Z

Lesson 7 Z-Scores and Probability

Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting raw score to Z-score to area Converting area to Z-score to raw score Introduction/Area Under the Curve Please note that area, proportion and probability are represented the same way (as a decimal value). Some examples require that you convert the decimal value to a percentage. Just move the decimal place to the right two places to turn the decimal into a percentage. Start this section by reviewing the first two topics in the above outline on the web page. Find the Z-score animated demonstrations or click here http://faculty.uncfsu.edu/dwallace/sz-score.html Using the Z-table Knowing the number of standard deviations from the mean gives us a reliable way to know how likely a score is for a population. There is a table of z-scores that gives the corresponding areas or probabilities under the curve. You will need the z-table in Appendix B of your text for this discussion. See page A-24 through A-26 in your text. The table shows the z-score in the left column, and then the proportion or area under the curve in the body, and finally there is a third column that shows the proportion or area under the curve in the tail of the distribution. Whenever we compute a z-score it will fall on the distribution at some point. The larger portion is the body, and the smaller portion is the tail.

If the z-score is positive, then the body will be the area that is below that z-score, and the tail will be the area that is above that z-score. If the z-score is negative, then the body will be the area that is above that z-score, and the tail will be the area that is below that z-score.

Converting a Z-Score to an Area Finding areas below a z-score What area lies below a z-score of +1? If we look this z-score up on the table we find that the area in the body is .8413 and the area in the tail is.1587. Since the z-score is positive and we want the area below the z-score, then we will want to look at the body area. So, .843 is the proportion in the population that have a z = 1 or less.

Now lets consider the situation if the z-score is negative. What area lies below a z-score of -1.5? You will not find negative values on the table. The distribution is symmetric, so if we want to know the area for a negative value we just look up the positive z-score to find the area. We will use a different column on the table, and that is why we must consider whether z is positive or negative when using the table. If we look this z-score (1.5) up on the table we find that the area in the body is .9332, and the area in the tail is .0668. Since the z-score is negative and we want the area below that point we will be using the tail area. So, .0668 is the proportion in the population below a z-score of -1.5.

Finding areas above a z-score The process for this type of problem is the same as what we have already learned. The only difference is in which column we will be using to answer the question. What area lies above a z-score of +1? If we look this z-score up on the table we find that the area in the body is .8413 and the area in the tail is.1587. Since the z-score is positive and we want the area above the z-score, then we will want to look at the tail area. So, .1587 is the proportion in the population that have a z = 1 or more. Now lets consider the situation if the z-score is negative. What area lies above a z-score of -1.5? Again, you will need to look up the positive z-value for 1.5. If we look this z-score up on the table we find that the area in the body is .9332, and the area in the tail is .0668. Since the z-score is negative and we want the area above that point we will be using the body area. So, .9332 is the proportion in the population above a z-score of -1.5. Finding areas between two z-scores When we have two different z-scores and want to find the area between them, we first must consider if both values are on the same side of the mean, or if one value is positive and the other negative. For our table, if the values are either both positive zs or both negative zs, we can find the tail area for both z-scores and subtract the two areas. You could just as easily find the two body areas for both z-scores and subtract them as well. For example, what is the area between Z = 1 and Z = 1.5? Since both scores are positive and we want the area between them, we will look up the tail area and subtract the two table values. Note that you never subtract z-scores, only areas from the table.

On the other hand, if you have one positive and one negative z-score then you must use the body area for either one of the z-scores, and the tail area for the other. Once you get the two areas off the table, then you subtract the two areas. For example, what is the area between Z = -1 and Z = 1.5? Since one score is positive and the other negative, the area we are looking for will cross the mean. Use the body area for one value and the tail area for the other. Once you get these values off the table, subtract them to find the area in between.

Converting a Raw Score to a Z-score and then into an Area These problems are exactly the same as the others we have been working. You must still find areas above/below/and between two z-scores, but now you must first compute the z-value using the z-formula before using the table. For example lets look at IQ scores for the population with a mean of 100 and standard deviation of 15 ( = 100, = 15). What proportion of the population will have an IQ of 115 or less? I first must compute Z. It is equal to z = 1. Now the question becomes what proportion of scores lie above z = 1?

Please review other examples of this type of problem on the web-page. Find the link to Z-scores and probability or click here http://faculty.uncfsu.edu/dwallace/sz-score2.html Converting an Area to a Z-score and then into a Raw Score For these problems we will be doing the same process we have been doing, but everything will be done in reverse order. We will start with a given area or proportion. You then use the z-table to find the area. However, when you use the z-table for these problems you must look up the area in either the body or tail column and then trace it back to find the z-score. Once we get the z-score we will plug in the values we know and solve for X in the z-score equation. For example, IQ scores for the population with a mean of 100 and standard deviation of 15 ( = 100, = 15), what score cuts off the top 10% of the distribution? Notice that these questions are always asking what the score is for a certain point. We are solving for X now. Prior examples were all asking for an area or proportion. The first step to solving this type of problem is to find the Z-score. We wont be computing z, but instead finding it from the table. Since we want the top 10%, we will be looking for the area on the table where the tail is .10 and the body is .90. You can look in either column, and it might help to draw the distribution in order to be sure you are using the right column. Make sure you are not using the z-column at this point.

We find the z-score that leaves the tail at the top of the distribution equal to .10 is Z = 1.28. Always use the z-score closest to the area of interest. Once we have that number,

we can plug in what we know into the z-formula and solve for X. Alternatively, if you have trouble with algebra, you can use the following formula: X = Z +

Special note for values in the lower 50% of the distribution: Whenever we want to find a z-score for a value below the mean, we must remember to make the value negative. Recall that the z-table only gives positive z-values. If the value is below the mean, then you must remember to insert the negative sign before doing the computation. For example, if I were looking for the IQ score for the bottom 10% of the distribution, in the above example, then I would have look up a tail area of .10 or body area of .90. The z-score we need is -1.28 even though the table shows only the positive value. Please review other examples of this type of problem on the web-page. Find the link to Z-scores and probability or click here http://faculty.uncfsu.edu/dwallace/sz-score2.html

Lesson 8 Probability

Outline Probability of an Event Probability of Single Events Probability of Multiple Events -without replacement -mutually exclusive events Conditional Probability Probability of an Event There are three classes of events: Impossible Events--------------Possible Events-----------Certain Events P = 0 P = 0 to 1 P = 1 The probability (P) of an impossible event is zero because there is zero chance of it happening. A certain event has a probability of one or 100% because it will always happen. Most of the events we will be interested in are possible events. These probabilities will always have a value between zero and one. Always leave your answer in decimal form, instead of fractional form. Simply divide out any fraction you have by dividing the top number by the bottom number. So, is 0.25. A simple experiment we could run to examine probabilities is to roll a six-sided die. What is the probability of rolling a 5 on a single die roll? Most people know it is 1/6 or .167 because there is only one side that is a 5 and there are six sides on the die. You can determine the probability of any even in this manner.

Probability itemsTotalcriteriainitemsP

__#___#)( =

So, we were only looking for one side in the last problem, out of a total of six sides. Another experiment we could use to look at probabilities is drawing cards from a standard deck. If I draw a card from a deck of cards what is the probability it is a heart?

P 25.041

5213)( ===Heart

Note that a standard deck of cards has 52 cards with 13 hearts/13 clubs/13 diamonds/13 spades. Since diamonds and hearts are red cards and the rest are black, there are 26 red and 26 black cards.

Probability of Single Events An individual event is a single event. With single events we are measuring the likelihood of a one thing happening. We might be interested in different outcomes, but we are still just going to roll the die once or draw a single card from a deck. For example, what is the probability that I roll a 5 or a 6 on a single die roll or P (5 or 6)? With single events you will see this or connector and you will add the two individual probabilities. So: P (5 and 6) = 1/6 + 1/6 = .167 + .167 = .334 What is the probability I draw a Heart or Club with a single draw from a standard deck of cards? P (Heart or Club) P(Heart or Club) = P(Heart) + P(Club) = 13/52 + 13/52 = .25 + .25 = .5 Probability of Multiple Events With multiple events we will be interested more than one outcome be realized. So, we will roll the die more than once or draw more than one card from a deck. For example, what is the probability of rolling a 5 and a 6 on two die rolls. To get both a five and a six I will have to roll the die more than once. When you see this and connector you will multiply individual probabilities. P (5 and 6) = P(5) * P(6) = 1/6 * 1/6 = .167 * .167 = .028 We will only be dealing with independent events in this section, or events that do not affect the outcome of other events. With the card experiment, then, we will not look at multiple draws where one draw could affect the probability of a separate event. For example, what is the probability of drawing a Heart and a Club from standard deck? P(Heart and Club) = P(Heart) * P(Club) = 13/52 * 13/52 = .25 * .25 = .062

Without Replacement Although we will focus on independent events like the last example, we will also consider what happens to probabilities in situations in which there is no replacement. The above examples assumed that once we drew a card from the deck that it was replaced before another draw was made. Notice that when figuring how many total events there were we used 52 every time because we assumed each draw was from a fresh deck. If the problem, however, specifies that there is no replacement then we must take this into account when figuring the probabilities. For example, what is the probability of drawing a Heart and a Club from a deck without replacement? When we count how many cards are left for the Club draw, there will be one less card in the deck because we already had to draw the Heart from the deck. Thus: P(Heart and Club) = P (Heart) * P (Club) = 13/52 * 13/51 = .25 * .255 = .064 We might also have to subtract a value from the numerator as well as the denominator. Try to find the probability of drawing three red cards from a deck without replacement. (Answer: 0.1176) Mutually Exclusive Events Mutually exclusive events are events that cannot happen together. For example, being a freshman and a sophomore are mutually exclusive. You are either one or the other but not both. For mutually exclusive events the probability the two events will occur together must always equal zero. Conditional Probability With conditional probabilities we will consider the probability of an event given that some other event has already happened. Thus, these are not independent events, and the rules we learned above will not apply. For these problems frequency data (or counts) will be given in a contingency table. This table will display the frequencies for different combinations of events. For example, consider the probability of having a computer or not, and living the U.S. or elsewhere. In U.S. Not in U.S. Computer 30 15 No Computer 10 20

Before we consider conditional probabilities, lets look at some of the types of questions we have already examined. For example, what is the probability that choosing someone from our sample will yield a person with a computer? To answer this question we will need to add up the total for each row and column in the table: In U.S. Not in U.S. Total Computer 30 15 45 No Computer 10 20 30 total 40 35 Since there are a total of 45 people in our sample with a computer out of 75 total people, there is a 0.6 probability that a random draw will yield a person with a computer. Now find the probability that a random draw will yield someone with a computer that is living in the U.S. Instead of looking in the total column for this type of problem, we will use one of the original values. There are 30 people living in the U.S. that also have a computer. So, 30 out of the total of 75 people or 0.4 live in the U.S. and have a computer. For a single event in a table like this one, use the values in the margin or the totals, and divide by the total number in the sample. For a combined event, use the original table values out of the total. For conditional probabilities we will restrict our sample to those items given to have already have happened. For example we might know the probability of having a computer and living in the U.S. for the entire sample. We could make this conditional by saying what is the probability of having a computer given that we know the person is living in the U.S.? With the second question we are not asking the probability of picking a person at random from the total, but instead we are restricting our sample to just those that live in the U.S. For conditional probabilities the total or denominator is the value given. For this example it is the total for those living in the U.S. or 40. We want to know what proportion have a computer out of these 40 people living in the U.S. Since 30 of those in the U.S. have a computer out of 40 the probability is 30/40 = .75 In this same example what is the probability someone does not live in the U.S. given they have a computer? We can write: P (not in U.S. | computer) Where the first probability is the one we are interested in, the vertical line means given and the second probability computer is what is given. Again, our new total is those with a computer or 45. We want to know the proportion out of these that are not in the U.S. or 15. So, the conditional probability is 15/45 = .33

Lesson 9 Hypothesis Testing

Outline Logic for Hypothesis Testing Critical Value Alpha () -level .05 -level .01 One-Tail versus Two-Tail Tests -critical values for both alpha levels Logic for Hypothesis Testing Anytime we want to make comparative statements, such as saying one treatment is better than another, we do it through hypothesis testing. Hypothesis testing begins the section of the course concerned with inferential statistics. Recall that inferential statistics is the branch of statistics in which we make inferences about populations from samples. Up to this point we have been mainly concerned with describing our distributions using descriptive statistics. Hypothesis testing is all about populations. Although we will start using just one value from the population and eventually a sample of values in order to test hypotheses, keep in mind that we will be inferring that what we observe with our sample is true for our population. We will want to see if a value or sample comes from a known population. That is, if I were to give a new cancer treatment to a group of patients, I would want to know if their survival rate, for example, was different than the survival rate of those who do not receive the new treatment. What we are testing then is whether the sample patients who receive the new treatment come from the population we already know about (cancer patients without the treatment). Again, even though we are talking about a sample, we infer that the sample is just part of an entire population. The population is either the one we already know about, or some new population (created by the new treatment in this example). Logic 1) To determine if a value is from a known population, start by converting the value to a z-score and find out how likely the score is for the known population. 2) If the value is likely for the known population then it is likely that it comes from the known population (the treatment had no effect). 3) If the value is unlikely for the known population then it is probably does not come from the population we know about, but instead comes from some other unknown population (the treatment had an effect).

4) A value is unlikely if it is less than 5% likely for the known population. Any value that occurs 5% or more of the time for the known population is likely and part of the known population. The 5% cut-off point is rather arbitrary, and it will change as we progress. For now we will use it as a starting point to illustrate several concepts.

Lets look at a simple example. Say the earth has been invaded by aliens that look just like humans. The only way to tell them apart from humans is to give them an IQ test since they are quite a bit smarter than the average human. Lets say the average human IQ (the known population) is = 100 = 15. We want to know if Bob is an alien. Bob has an IQ score of 115? Is it likely that he comes from the known population and is human, or does he come from a different alien population? To answer the question, first compute the z-score.

11515

15100115 ===Z Next, find out how likely this z-score is for the population.

For hypothesis testing we will always be interested in whether the value is extreme for the population, or unlikely. Thus, we will be looking in the Tail Column when deciding if the value is unlikely.

So, the likelihood of observing an IQ of 115 or more is .1587. Since the probability is not less than 5% or .05 we have to assume Bob comes from the general population of humans. Say Neil has an IQ of X = 30. Is it likely Neil comes from the general population? Again we first compute the z-score, and then find how likely it is to get that value or one more extreme.

21530

15100130 ===Z Next, find out how likely this z-score is for the population.

So, the likelihood of observing an IQ of 130 or more is .0228. Since the probability is less than 5% or .05 we have to assume Neil comes from a different population than the one we know about (the general population of humans). Critical Value Critical Values are a way to save time with hypothesis testing. We dont really have to look up the probability of getting a particular value in order to verify it is less than 5% likely. The reason for this fact is that the z-score that marks the point where a value becomes unlikely does not change on the z-scale. That is, there is only one z-score at the that is 5% likely. Any z-score beyond that point is less than 5% likely. Thus, I dont have to look up each particular area when I compute my z-score. Instead I only have to verify that the z-score I computed is more extreme than the one that is 5% likely. So, we can stop once we compute the z-score without reference to the z-table. What z-score will be exactly 5% likely for any population? This is the z-score we will make comparisons against. Use the z-table to determine the z-score that cuts off the top 5% of scores. Finding the critical value is important, and will be one of the steps that must be performed anytime we conduct a hypothesis test. Since we will be doing hypothesis testing from this point on, many points on subsequent exams will come from just knowing the critical value.

So, Z = 1.64 is the critical value. Any value more extreme than 1.64 is unlikely, and all other events will be likely for the known population. If Z=2.1, or Z=1.88 you conclude that the value is unlikely, and so must be part of a different population. If Z = 1.5, or Z =0.43 you conclude that the value is likely, and so must be part of the known population. Note that we were only working on one side of the distribution in the above problem. If we were interested in a value that was below the mean, instead of above it, then we would flip our decision line to the other side. Since the distribution is symmetric, the numbers will not change.

Alpha Alpha is the probability level we set before we say a value is unlikely for a known population. The critical value we just found is only one that we will use. It assumes that a value must be less than 5% likely to be unlikely, and therefore part of a different population. Alpha was .05 ( = .05) for that example. Sometimes researchers want to be very sure before they decide a value is different. Thus, we will also use an alpha level of .01 or 1% as well. If alpha is .01, then a value must be less than 1% likely before it is said to be unlikely for a known population. If alpha is 1% then the critical value will be different than the one we found above. Alpha is given in every problem, but you must use that information to determine the critical value. What z-score will be exactly 1% likely for any population? This is the z-score we will make comparisons against when alpha is set to 1% ( = .01). Use the z-table to determine the z-score that cuts off the top 1% of scores just like the last example. Use the Tail column and find .01. You could also look in the Body Column and find .99. We

will use Z = 2.33 when Alpha is set to the 1% level. Also note that when we are interested in determining if values below the mean are unlikely our critical value will be negative. One-Tail versus Two-Tail Tests Another factor that will affect our critical value is whether we are performing a one or a two-tail test. The critical values we have looked at so far were for one-tail tests because we were only looking at one tail of the distribution at a time (either on the positive side above the mean or the negative side below the mean). With two-tail tests we will look for unlikely events on both sides of the mean (above and below) at the same time. I will discuss how to determine if a problem is a one or a two tail test in a later lesson, but lets go ahead and find the critical values for two-tailed test the same way we did the one-tail tests above. Lets begin with an alpha level of 5%. We still want 5% of our events to be unlikely and 95% of our events to be likely for the known population. Now, however, we want to be looking for unlikely events in both directions at the same time. So, we will split the unlikely block into two parts, each half the total 5% area.

What z-scores will then mark the middle 95% of our distribution? You will have to look up an area of .025 in the Tail Column of the z-table.

The process for finding the two-tail critical values when alpha is set to .01 is the same. This time we will want 99% of our values in the middle, leaving only .005 or half of one percent on each side. Can you find the critical value on the z-table (answer: z = 2.58).

So, we have learned four critical values. 1-tail 2-tail = .05 1.64 1.96/-1.96 = .01 2.33 2.58/-2.58 Notice that you have two critical values for a 2-tail test, both positive and negative. You will have only one critical value for a one-tail test (which could be negative).

Lesson 10 Steps in Hypothesis Testing

Outline Writing Hypotheses -research (H1) -null (H0) -in symbols Steps in Hypothesis Testing -step1: write the hypotheses -step2: find critical value -step3: conduct the test -step4: make a decision about the null -step5: write a conclusion Writing Hypotheses Before we can start testing hypotheses, we must first write the hypotheses in a formal way. We will be writing two hypotheses: the research (H1) and the null (H0) hypothesis. The research hypothesis matches what the researcher is trying to show is true in the problem. The null is a competing hypothesis. Although we would like to directly test the research hypothesis, we actually test the null. If we disprove the null, then we indirectly support the research hypotheses since it competes directly with the null. We will discuss this fact in more detail later in the lesson. Again, the research hypothesis matches the research question in the problem. Lets take a look at a sample problem: Suppose some species of plants grows at 2.3 cm per week with a standard deviation of 0.3 ( = 2.3 = 0.3). I take a sample plant and genetically alter it to grow faster. The new plant grows at 3.2 cm per week (X = 3.2). Did the genetic alteration cause the plant to grow faster than the general population? Set alpha = .05. Lets focus on writing hypotheses, rather than any other steps we have learned for now. In order to write the research hypothesis look at what the researcher is trying to prove. Here we are trying to show that the genetically altered plant grows at a faster rate than unaltered plants. Thats what we want the research hypothesis to say. However, when you write your hypotheses, be sure to include three elements: 1) explicitly state the populations you wish to compare. For now, one will be a treatment population and the other will always be the general population. 2) State the dependent variable. We have to be explicit about the scale on which we expect to find differences. 3) State the type or direction of the effect. Are we predicting the treatment population will be greater or less than the general population (1-tail)? Or, are we looking for differences in either direction at the same time (2-tail)? The above problem is one-tail since we are looking for a growth rate higher than the average. Look for words that indicate a direction in the problem for one-tail test (e.g. higher/lower, more/less,

better/worse). It would be two-tailed if the problem had stated that we expected a different growth rate than the general population. Different could be higher or it could be different because it is lower. The current example is easy to translate into a hypothesis, but check the homework packet because the wording is not always so obvious. For the research hypotheses (denoted by H1 ): H1: The population of genetically altered plants grows faster than the general population. You could vary the wording a bit, as long as you include the three elements. Notice that we state both the treatment population and the population we will compare that to, the general population. Growth rate is the dependent variable, and we indicate the direction by saying it will grow faster. The null hypothesis (denoted by H0) is a competing hypothesis. Its basically the opposite of the research hypothesis. In general it states that there is not effect for our treatment or no differences in our populations. For this example: H0: The population of genetically altered plants grows at the same or lower rate as the general population. Ive included the same or lower wording for the one-tail test because we want to cover all the possible outcomes of the test. We only want to show that the treatment population grows faster. If they end up growing slower it wont support the research hypothesis, so we include left-over elements with the null. For two-tail tests, substitute different for the word faster in the research hypothesis. The two-tail null would say the groups are do not differ. In Symbols We can also write the hypothesis in notational form. We will restate both the null and research hypotheses in symbols we have been using for our formulas. Thus: H1: gen.alt. > 2.3 H0: gen.alt < 2.3 Notice that we represent the treatment population with a mu (). We do this because we want to make inferences about the population, not the single value sample I am using to test the hypothesis. Our inferences will be that the entire population the plant comes from grows at a faster rate. The value of 2.3 is the general population mean we are comparing against. Although it is represented with a mu in the problem, we dont the

symbol because we know the exact value for that population. For two-tail test we simply change the direction arrows to equal/not-equal signs (an = sign for the null and / sign for the research hypothesis). Steps in Hypothesis Testing Now we can put what we have learned together to complete a hypothesis test. The steps will remain the same for each subsequent statistic we learn, so it is important to understand how one step follows from another now. Lets continue with the example we have already started: Suppose some species of plants grows at 2.3 cm per week with a standard deviation of 0.3 ( = 2.3 = 0.3). I take a sample plant and genetically alter it to grow faster. The new plant grows at 3.2 cm per week (X = 3.2). Did the genetic alteration cause the plant to grow faster than the general population? Set alpha = .05. Step 1: Write the hypotheses in words and symbols H1: The population of genetically altered plants grows faster than the general population. H0: The population of genetically altered plants grows at the same or lower rate as the general population. H1: gen.alt. > 2.3 H0: gen.alt < 2.3 Step 2: Find the critical value for the test Since alpha is .05, and it is a one-tail test because we think our treatment will produce plants that grow faster than the general population: Zcritical =1.64 Step 3: Run the test Here we find out how likely the value is by computing the z-score.

33.09.0

3.03.22.3 ===Z

Step 4: Make a decision about the Null Reject the Null or Fail to Reject the Null (retain the null) are the only two possible answers here. Since the value we computed for the z-test is more extreme than the

critical value, we reject the Null. Graphically, though not required for the answer, we have:

Note that we are testing the Null. It is either proven or disproven. We never prove the research hypothesis, even thought that is our intent. Instead, if we disprove the null, we indirectly support the research hypothesis. This is true, because you will notice that our decision is based on the statistical test that the treatment value is not likely to have come from the same population. We infer it is a different population, but we actually prove that it is not from the same population. It may seem like a matter of semantics, but indulge me on this one. Step 5: Write a conclusion For this example, we conclude: The population of genetically altered plants grows at a different rate than the general population. Although we have a conclusion in step 4, write a conclusion here in plain language without any statistical jargon. What did our test show? If you reject the null, then the then there was a difference (treatment had an effect). The research hypothesis is your conclusion (you can simply restate it from Step 1). If you fail to reject the null, then the null hypothesis is your conclusion (again, you can just rewrite it from Step 1).

Lesson 11 Hypothesis Testing with a Sample of Values

We have looked at the basics of hypothesis testing using the z-formula we had already learned. However, we never test a hypothesis based on one individual from a population. Instead, we will want to have a sample of values to test against the population. The formula we will want to use has a minor change from the one we have been using.

x

Xz = , where

x = n

Notice that there is sample mean now in the numerator instead of just a single x-value. Often this will be given just like the x-value in prior problems, but now you may also have to compute it from the sample. Compute x first for the denominator by dividing the standard deviation by the square root of the given sample size (n). Once you get that number plug it in as the denominator in the z-score formula. The rest of this lesson is devoted to the theory behind the changes we make when moving from tests with a single x-value to tests with samples of x-values. There are no computational additions for the exam other than the formula change above. However, you should be concerned with understanding the conceptual meaning of this lesson. At a minimum you should be able to recognize the rules of the Central Limit Theorem for the exam (detailed below).

The lesson continues on the web page. Take notes on the sampling distribution page, the standard error page, and the standard error with hypothesis testing page. Links to these pages are provided below. It is important to review each one. Again, these lessons contain conceptual information for the most part, however, the last page is devoted to the computations you will perform for the exams.

Sampling Distributions

Standard Error

Standard Error and Z-score Hypothesis Testing

Lesson 12 Errors in Hypothesis Testing

Outline Type I error Type II error Power Examples in the real world Anytime we make a decision about the null it is based on a probability. Recall that we reject the null when it tests a value that is unlikely for the known population. Extreme values are unlikely for the population, but not impossible. So, there is always some chance that our decision is in error. Note that we will never know whether we know we have made an error or not with our hypothesis test. When running a test, I only know what my decision is about the test, and not the true state of reality. Thus, this discussion on errors is strictly theoretical. Type I Errors Whenever a value is less than 5% likely for the known population, we reject the null, and say the value comes from some other population.

Notice that we are saying the value is really from another population distribution out there that we dont know about.

However, some of the time the value really does come from the known population. Notice that even though the value represented is beyond the critical value it still lies under the curve for the normal population. We reject any values in this range, even though they really are part of the known population. When we reject the null, but the value really does come from the known population a Type I error has been committed. A Type I error, then, happens when we reject the null when we really should have retained it. Note that a Type I error can only occur when we reject the null. The part of the distribution that remains under the curve for the known population but is beyond our critical value in the region of rejections is alpha (). When we set alpha we are setting the probability of making a Type I error.

Type II Errors Whenever a value is more than 5% likely for the known population, we retain the null, and say the value comes from the known population. But, some of the time the value really does come from a different unknown population.

Notice in this situation the value is below the critical value, so we retain the null. However, the value is still under the unknown population distribution, and may in fact come from the unknown population instead. Thus, when I retain the null, when I should really have rejected it I commit a Type II error. The probability of making a Type II error is equal to beta and not strictly defined by alpha. Although we know the probability of a Type I error because we set alpha, a Type II error takes in a few more factors than that. You can see the region of Beta () below. Notice that it is the area below the critical value, but that is still part of the other unknown distribution.

Power Power is the probability of correctly rejecting the null hypothesis. That is, it is the probability of rejecting the null when it is really false. Again, we never really know if the null is false or not in reality. Power is another way of talking about Type II errors. Such errors have been recognized as a problem in the behavioral sciences, so it is important to be aware of such concepts. However, we will not be computing power in this course. You can ignore the power demonstration on the web page for that reason. An easy way to remember all these concepts might be to put them in a table, much like your textbook does.

Examples of Errors in the Real World Another way to think about Type I and Type II errors is to think of them in terms of false positives and false negatives. A Type I error is a false positive, and a Type II error is a false negative. A false positive is when a test is performed and shows an effect, when in fact there is none. For example, if a doctor told you that you were pregnant, but you were not then it would be a false positive result. The test shows a positive result (what you looking for is there), but the test if false. A false negative is when a test is performed and shows no effect, when in fact there is an effect. The opposite situation of the above example would apply. A doctor tells you that you are not pregnant, when if fact you are pregnant. The test shows a negative result (what you are looking for is not there), but the test is false. Lets look at another example. A sober man fails a blood alcohol test. What type of error has been committed (if any)? For this type of problem you will get two pieces of information. First, whether the test was positive or negative. The test is positive if what you are looking for is found. It is negative if the test shows what you are looking for is not there. The second piece of information is whether the test is in error or not (false or true test). Thus, for this example, the test is positive because if you fail a blood alcohol test it is showing that there is alcohol in your system. You are positive for alcohol in that case. Since the man is sober, the test is false. So, here we have a false positive test or Type I error.

Lesson 13 Hypothesis Testing with the t-test Statistic

Outline Unknown Population Values The t-distribution -t-table Confidence Intervals Unknown Population Values When we are testing a hypothesis we usually dont know parameters from the population. That is, we dont know the mean and standard deviation of an entire population most of the time. So, the t-test is exactly like the z-test computationally, but instead of using the standard deviation from the population we use the standard deviation from the sample.

The formula is: t = X

sx , where sx = sn

The standard deviation from the sample (S), when used to estimate a population in this way, is computed differently than the standard deviation from the population. Recall that the sample standard deviation is S and is computed with n-1 in the denominator (see prior lesson). Most of the time you will be given this value, but in the homework packet there are problems where you must compute it yourself. The t-distribution There are several conceptual differences when the statistic uses the standard deviation from the sample instead of the population. When we use the sample to estimate the population it will be much smaller than the population. Because of this fact the distribution will not be as regular or normal in shape. It will tend to be flatter and more spread out than population distribution, and so are not as normal in shape as a larger set of values would yield. In fact, the t-distribution is a family of distributions (like the z-distribution), that vary as a function of sample size. The larger the sample size the more normal in shape the distribution will be. Thus, the critical value that cuts off 5% of the distribution will be different than on the z-score. Since the distribution is more spread out, a higher value on the scale will be needed to cut off just 5% of the distribution. The practical results of doing a t-test is that 1) there is a difference in the formula notation, and 2) the critical values will vary depending on the size of the sample we are using. Thus, all the steps you have already learned stay the same, but when you see that the problem gives the standard deviation from the sample (S) instead of the population (), you write the formula with t instead of z, and you use a different table to find the critical value. The t-table Critical values for the t-test will vary depending on the sample size we are using, and as usual whether it is one-tail or two-tail, and due to the alpha level. These critical values

are in the Appendices in the back of your book. See page A27 in your text. Notice that we have one and two-tail columns at the top and degrees of freedom (df) down the side. Degrees of freedom are a way of accounting for the sample size. For this test df = n 1. Cross index the correct column with the degrees of freedom you compute. Note that this is a table of critical values rather than a table of areas like the z-table. Also note, that as n approaches infinity, the t-distribution approaches the z-distribution. If you look at the bottom row (at the infinity symbol) you will see all the critical values for the z-test we learned on the last exam. Confidence Intervals If we reject the null with our hypothesis test, we can compute a confidence interval. Confidence intervals are a way to estimate the parameters of the unknown population. Since our decision to reject the null means that there are two populations instead of just the one we know about, confidence intervals give us an idea about the mean of the new unknown population. See the Confidence Interval demonstration on the web page or click here http://faculty.uncfsu.edu/dwallace/sci.html for the rest of the lesson.

Lesson 14 Independent Samples t-test

Outline No Population Values Changes in Hypotheses Changes if Formula -standard error Pooled Standard Error -weighted averages Critical Values -df Sample Problem No Population Values With the independent samples t-test we finally reach the point where we have no population values. This fact is important because when we test hypotheses we are usually testing an idea and a population that we know nothing about. Think about the kinds of scientific discoveries you hear about often. New treatments for diseases, new drugs, or new techniques for improving depression all involve testing a population created by the treatment or drug or technique. So, with the independent samples t-test we will compare two sample values directly. Note that we are still making the inference about the populations from which the samples are drawn. Changes in Hypotheses All hypotheses from this point on in the course will be two-tailed. In addition, since we no longer no any population values we will use mu to represent both populations. So for example, H0: diet = placebo H1: diet placebo Formula Changes

Recall the formula for the t-test we have been using: t = X

sx , where sx = sn

The numerator will now have two sample values )( 21 XX instead of one sample and one population. The denominator, recall, is the standard error (the standard deviation divided by the square root of the sample size).

Our standard error (denominator) was: sx = sn Remember that the standard error measures variability we expect to see among samples. Now that we have two samples we will want to include the estimate of variability fro