chapter 10 - 12 – significance/hypothesis testing · 10.2 – 10.4 [watch chapter 11 videos 1 –...

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Chapter 10 - 12 – Significance/Hypothesis Testing Lesson Objectives: In this chapter you will learn how to estimate the true proportion/mean of an entire population and how to test a claim about the proportion/mean of an entire population. Remember, you cannot gather information from every member of the population, so you collect information about a portion of the population (you take a sample). You could then use that sample to estimate the population proportion/mean or to test a claim that was made about the population proportion/mean.

Date Topics Objectives: Students will be able to: Assignments

Mar 25

10.2 The Reasoning of Significance Tests, Stating Hypotheses, Interpreting P-values, Statistical Significance

10.4 Type I and Type II Errors, Planning Studies: The Power of a Statistical Test

• State correct hypotheses for a significance test about a population proportion or mean.

• Interpret P-values in context. • Interpret a Type I error and a Type II error

in context, and give the consequences of each.

• Understand the relationship between the significance level of a test, P(Type II error), and power.

Read TPS Section 10.2 – 10.4

[Watch Chapter 11

Videos 1 – 4]

Mar 26

12.1 Carrying Out a Significance Test, The One-Sample z Test for a Proportion

12.2 Two-Sided Tests, Why Confidence Intervals Give More Information

• Check conditions for carrying out a test about a population proportion.

• If conditions are met, conduct a significance test about a population proportion.

• Use a confidence interval to draw a conclusion for a two-sided test about a population proportion.

Read TPS Section 12.1 – 12.2,

[Watch Chapter 11

Videos 5 – 8]

Mar 27

11.1 Carrying Out a Significance Test for 𝜇, The One Sample t Test, Two-Sided Tests and Confidence Intervals

11.2 Inference for Means: Paired Data, Using Tests Wisely

• Check conditions for carrying out a test about a population mean.

• If conditions are met, conduct a one-sample t test about a population mean 𝜇.

• Use a confidence interval to draw a conclusion for a two-sided test about a population mean.

• Recognize paired data and use one-sample t procedures to perform significance tests for such data.

Read TPS Section 11.1 – 11.2

[Watch Chapter 11

Videos 9 – 12]

Chapter 11 Packet Due April 5

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10.2 – Reasoning of Significance Tests & Stating Hypotheses [VIDEO #1] The two most common types of formal statistical inference are:

1. Confidence Intervals - ______________________________________________________________

2. Significance Tests - _______________________________________________________________

Both types of inference are based on the sampling distribution of statistics à Chapter 9 stuff!

In the following problems, determine if a Confidence Interval should be constructed or a Hypothesis Test carried out.

1. You measure the heights of a random sample of 400 high school sophomore males. The sample mean is 𝑥 = 66.2 inches. Suppose that the heights of all high school sophomore males follow a normal distribution with unknown mean µ and standard deviation σ = 4.1 inches. You desire to estimate the mean height for all high school sophomore males.

2. A researcher wishes to determine if students are able to complete a certain pencil-and-paper maze more quickly while listening to classical music.

3. An engineer designs an improved light bulb. The previous design had an average lifetime of 1200 hours. The mean lifetime of a random sample of 2000 of the new bulbs is found to be 1251 hours. Although the difference is small, he would like to know if the difference is statistically significant.

****************************************************************************************************************************** Just like CI’s might not capture the population parameter of interest (and we don’t know it), we might make a decision with a significance test that we believe is the right decision but ends up being the wrong one. Let’s make the claim p = 0.62 is the true proportion of CHS students who do not drink alcohol. Many of you believe that number is an overestimate, so you believe, in fact, p < 0.62. Let’s say we take a SRS of 100 students and ask them if they have drunk alcohol in the past 30 days. We find 𝑝 = 0.58 from our survey. Two things could happen as a result:

1. The initial claim (p = 0.62) is true, but because of sampling variability, we just happened to collect a sample proportion (𝑝 = 0.58) that is lower than the claim.

2. The initial claim (p = 0.62) is false, so the sample proportion (𝑝 = 0.58) was not unlikely to happen; thus proving our belief that, in fact, p < 0.62.

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Stating Hypotheses We will need to write {our claims} and {beliefs about our claims} in a more mathematical way. The initial claim you want to try to prove is wrong is the __________________________________, or ______.

• _______ will ALWAYS use _______________________________!!! The belief you want to prove is right is the _____________________________________________, or ______

• _______ will ALWAYS use one of the three _____________________________ ( ____ , ____ , ____ )

o If you use ______ or _______ for Ha, then it is called a ______________________ test.

o If you use ______ for Ha, then it is called a ________________________ test. Hypotheses ALWAYS refer to a ___________________________, never to a ____________________. In addition to writing the two hypotheses, you MUST define the ___________________ of interest (or write what each hypothesis means to the problem…this takes more writing/effort!). Therefore, we would write our hypotheses using the previous example as: Ho: __________ ( _________________________________________________________________ ) Ha: __________ ( _________________________________________________________________ ) _______ = _________________________________________________________________________ Example: Zach Holland is an avid golfer who would like to improve his play. A friend suggests getting new clubs and lets Zach try out his 7-iron. Based on years of experience, Zach has established that the mean distance that balls travel when hit with his old 7-iron is µ = 175 yards with a standard deviation of σ = 15 yards. He is hoping that this new club will make his shots with a 7-iron more consistent (less variable), so he goes to the driving range and hits 50 shots with the new 7-iron.

(a) Describe the parameter of interest in this setting.

(b) State appropriate hypotheses for performing a significance test.

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10.3 – Interpreting P-values & Statistical Significance [VIDEO #2] Consider our judicial system. Arrested people are innocent until proven guilty. We could say the null hypothesis is “defendant is innocent” vs. the alternate hypothesis of “defendant is guilty”. Prosecutors must provide enough evidence against the defendant’s innocence to the jury for them to reject the null hypothesis in favor of the alternate hypothesis that the defendant is guilty of the crime. Do juries ever make the wrong decision? Of course! The two types of mistakes that juries could make will be discussed in the next video. For a jury to decide the defendant is guilty there needs to be a certain level of evidence for them to make that conclusion. If not enough evidence is provided to prove guilt, then the jury will fail to reject the defendant’s innocence. THIS IS A VERY IMPORTANT CONCEPT!!! à The jury NEVER PROVES the defendant is innocent, but rather they were not provided enough evidence to conclude the defendant was guilty. The amount of evidence needed to change the jury’s mind from innocent (failing to reject Ho) to guilty (rejecting Ho) is referred to as the ________________________________, or _____ level. This value is directly tied to the confidence level from confidence intervals from last chapter!!! A 95% CI = 5% α-level. A 99% CI = 1% α-level. Common α-levels are 1%, 5%, and 10%. The significance (α) value must be set BEFORE performing the significance test; otherwise, it’s cheating! In terms of a pole vaulter, if you know how high you can jump already, then you can set the bar low enough so you will always jump over it! If you set the bar first before you know how high you can jump, then you may or may not clear the jump. This last scenario is what we wish to do statistically…to be fair! The total evidence provided to the jury can be thought of as the _____________________. The definition of this term is “the probability, computed assuming Ho is true, that the statistic would take a value as extreme as or more extreme than the one actually observed is”. So, if the p-value (or total evidence assuming the defendant is innocent) is as extreme as or more extreme than the significance or α-level (that tipping point from assuming innocence to concluding guilty), we can reject Ho (innocence) and conclude Ha (guilt). This happens when the p-value ______________________ the α-level. If the p-value < α-level, we will say that the data are ______________________________________________ at the α-level. In that case, we _________________ the null hypothesis, Ho, and conclude that there IS convincing evidence in favor of the alternative hypothesis, Ha. In other words…

“IF THE P-VALUE IS TOO LOW (< α-level), THE ________________________!!!”

Example: Remember our golfer, Zach Holland, from the last video? When he took his data and performed a significance test at the 5% significance (α) level, he calculated a p-value of 0.0224.

(a) Interpret the results in context.

(b) Do the data provide convincing evidence against the null hypothesis? Explain.

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10.4 – Type I and II Errors [VIDEO #3]

• Tests of Significance assess the _____________ of evidence against the _______ hypothesis. We

measure evidence by the ____________, which is a probability computed under the assumptions that

_____ is true. The alternative hypothesis (the statement we seek evidence ______) enters the test only to

help us see what outcomes count against the null hypothesis.

• If our result is significant at level α, we reject ______ in favor of ______. Otherwise, we fail to reject H0.

Consider a large hospital that needs 50,000 baby needles of diameter 2mm. The shipment is in the right box,

but the needles look too big. The inspector is not going to measure every needle. He will randomly select a

sample and measure the diameters. Surely there will be a little variability in the measurements. Based on the

sample outcome, the shipment will either be accepted or rejected.

Type I and Type II Errors

• When we must make a decision based on our sample we have two choices:

1. Reject H0: We think the shipment of needles does not meet the standards

2. Fail to Reject H0: We think the shipment of needles meets standards

• There are two types of incorrect decisions:

1. We can reject a good shipment of needles – Waste of time for the manufacturer, and the hospital

still needs the needles so they must place another order

2. We can keep a bad shipment of needles – Injure the patients

Truth About the Population H0 True Ha True

Decision Based on

the Sample

Reject H0 Type I Error Correct Decision

Fail to Reject H0

Correct Decision Type II Error

Type I and Type II Errors:

If we reject H0 (accept Ha) when in fact H0 is true, this is a ______________________.

If we fail to reject H0 (reject Ha) when in fact Ha is true, this is a ___________________.

Type I and II Error Calculations:

1. Type I: Probability of Type I Error = ____ (more on this in the next video)

2. Type II: Probability of Type II Error = _____ (won’t calculate this in here)

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Type I and Type II Errors: Court Case

Truth About the Defendant Innocent Guilty

Jury’s Decision Based on

the Evidence

Reject Innocence

Type I Error “Innocent Found Guilty” (Bad for the Individual)

Correct Decision

Accept Innocence Correct Decision

Type II Error “Guilty Found Innocent”

(Bad for Society)

*************************************************************************************************************************

A new medication approved by the FDA has been accused of having harsh side effects for its patients. It was known that the medication may affect a patient’s kidneys. In order for kidneys to be considered “failing”, they must have lost 80% of their ability to function, thus resulting in dialysis. The pharmaceutical company who manufactures the drug claims that in their clinical studies, patients only experienced a kidney loss rate of 50% and therefore never required dialysis. A large number of patients have had to start dialysis treatment after being on the medication. A neutral, third-party company decides to settle the issue and takes a SRS of 500 patients currently on the drug to measure what percentage their kidneys are functioning at. 1. State the hypotheses and define the parameter of interest. 2. Describe Type I error and Type II error in this situation.

3. Which type of error is more serious? Why? 4. What significance (α) level do you think should be used in this situation? Why?

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10.4 – Type II Error & Power [VIDEO #4] 1. Type I Error: _____________________________________________________________________

The probability of Type I Error is just ____, the significance level of the test.

2. Type II Error: _____________________________________________________________________ Type II Error occurs when you fail to reject the null hypothesis, when you should reject it! That’s bad! The ____________________________________ of that is a new term called POWER! Power - ____________________________________________________________________________

• How can you increase the power of any hypothesis test?

o ______________________________________

o ______________________________________ Practice:

In the past, the mean score of the seniors at South High on the American College Testing (ACT) college entrance examination has been 20. This year a special preparation course is offered, and all 53 seniors planning to take the ACT test enroll in the course. The principal believes that the new course will improve the students’ ACT scores. A hypothesis was performed at the α = .01 level and found the power to be 0.68. a. State the hypotheses and define the parameter of interest. b. Describe the Type I and Type II error in this situation. c. Explain what the power of 0.68 means in this situation. d. Calculate the probability of Type I Error and Type II Error. e. How could you increase the power of the test?

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12.1 – Carrying out a Significance Test for p [VIDEO #5]

In addition to performing the same “three C’s” we did for CI’s, you must do what?

__________________________________________________________________________________________

“C”onditions: 1. The data must come from a random sample from the population.

a. YOU MUST WRITE BEFORE DOING ANY CALCULATIONS: “We have a SRS of (size/#)

(people or whatever you sampled) to represent ALL (people or whatever you sampled).

b. If there is no mention of a random sample, then write “Sample not known to be random,

proceed with caution!” and continue with the problem.

i. IF data came from a sampling method of high bias (like volunteer response), then

mention this information and STOP. It’s not worth calculating & concluding anything. ii. IF data came from a sampling method not necessarily associated with high bias (like a

convenience sample), then mention this and say “Proceed with caution!”

2. The observations of your sample must be independent. a. Check the “10%” condition a.k.a. “Population > 10·n”. SHOW YOUR WORK!

If it checks, then write “We may calculate the standard error.”

b. If NOT, then you need to use the Finite Population Correction formula we mentioned before but will not use in this class, so write “Standard error cannot be safely calculated.” STOP.

3. The sampling distribution of 𝑝 must be approximately normal. a. Check the condition n· 𝑝! > 10 and n(1 - 𝑝!) > 10 where 𝑝! is the hypothesized value for p.

SHOW THE ACTUAL WORK HERE!

If it checks, then write “We may assume the sampling distribution is approx. normal.”

b. If NOT, then write “Sampling distribution not normal, hypothesis test cannot be

performed.” STOP.

c. NOTE: The CLT does NOT work with proportion problems, so don’t use it here!!! “C”alculations: In video #2, we discussed the concept of a p-value (total evidence against the defendant) in comparison to the significance level (the tipping point where the jury goes from thinking the defendant is not guilty to guilty). If the p-value was smaller (or more extreme than) the significance level, we rejected Ho in favor of Ha. We now need to discuss how to calculate that p-value! To get the p-value we need a test statistic.

We will get into the specifics of this formula in the next video, but just know this for now…

The “test statistic” is really a _____________ that says how “far away” our actual sample value is from our hypothesized population value. If it’s “far enough” away, then Ho is more than likely wrong!

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When we did CI’s, I told you to not worry about the negative you would get with invNorm or invT, but you DO need to keep whatever sign you get when calculating the test statistic! We no longer just “±” it!!! Now, the z-score is from a _________________________________________________________________, which has a mean of ______ and standard deviation of ________. The p-value is the area under the curve “as extreme as or more extreme than” this z-score (test statistic). Remember how to find the area under the curve with lower and upper bounds??? _______________________ Draw a normal curve, find the approximate spot for the z-score, and then shade…WHICH WAY??? The shading depends on the ___________________________________________. If it uses ____, then shade to the ___________ of the z-score. If it uses _____, then shade to the __________ of the z-score. If it uses _____, then we will shade to the outside of both the positive and negative version of the z-score. Thus, we will need to _________________ our p-value in the end! “C”onclusion: And if the p-value is ____ our pre-determined α-level, then we get to reject Ho in favor of Ha!!! Example You want to test the notion of “home-court advantage”, that the home team wins more than half of all games. Out of 30 NBA games between January 3rd and 7th 1992, 18 were won by the home team. We will assume all conditions have been met. The standard deviation of the statistic is 0.09129 (you will calculate this value directly in the next video). State the null and alternate hypotheses, calculate the test statistic and p-value. What would you conclude at the 5% significance level?

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12.1 – One-sided z-Test for p [VIDEO #6] Why is this entitled “one-sided”??? When the alternate hypothesis uses ____ or ____, then we are only considering ONE SIDE of the test statistic (z-score). We will discuss a “two-sided” test in the next video!!!







mention this information and STOP. It’s not worth calculating & concluding anything.

ii. IF data came from a sampling method not necessarily associated with high bias (like a

convenience sample), then mention this and say “Proceed with caution!”

2. The observations of your sample must be independent.

a. Check the “10%” condition a.k.a. “Population > 10·n”. SHOW YOUR WORK! If it checks, then write “We may calculate the standard error.”

b. If NOT, then you need to use the Finite Population Correction formula we mentioned before but

will not use in this class, so write “Standard error cannot be safely calculated.” STOP.

3. The sampling distribution of 𝑝 must be approximately normal.

a. Check the condition n · 𝑝! > 10 and n(1 - 𝑝! ) > 10 where 𝑝! is the hypothesized value for p. SHOW THE ACTUAL WORK HERE!

If it checks, then write “We may assume the sampling distribution is approx. normal.”

b. If NOT, then write “Sampling distribution not normal, hypothesis test cannot be

performed.” STOP.

c. NOTE: The CLT does NOT work with proportion problems, so don’t use it here!!! “C”alculations: In the last video, we discussed the general formula to calculate the test statistic:

which will really turn into…

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The p-value is the area under the curve “as extreme as or more extreme than” this z-score (test statistic). Remember how to find the area under the curve with lower and upper bounds??? _______________________ “C”onclusion: And if the p-value is ____ our pre-determined α-level, then we get to reject Ho in favor of Ha!!!

• Here’s what you write as your concluding statement: “We (circle one à ) do / do not have enough

evidence at the (α) level to conclude that the (population parameter in context) for all (what/who

is the population in context) is (state the alternate hypothesis here).”

• The conclusion statement ALWAYS refers to the alternate hypothesis, ALWAYS!!!

Example In the case of Hazelwood School District v. United States (1977), the U.S. Government used the City of Hazelwood, a suburb of St. Louis, on the grounds that it discriminated against blacks in its hiring of school teachers. The statistical evidence introduced notes that of the 405 teachers hired in 1972 and 1973 (the years following the passage of the Civil Rights Act), only 15 had been black. The proportion of black teachers living in the county of St. Louis at the time was 5.7% if one does not include the city of St. Louis. State the hypotheses & calculate the test statistic and p-value. What conclusion would you make at the 5% significance level?

Question: Do I have to write down the test statistic (z-score) to receive full credit on problems when all we really need and use is the p-value to determine if we reject the null hypothesis??

Answer: YES. This is required on the AP Stats exam, so I will require you to write it down, too!

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12.1 – Two-sided z-Test for p [VIDEO #7]

2-Sided Test 1-Sided Test (Lower) 1-Sided Test (Upper)

1. State Hypothesis

Null Hypothesis H0: p = ____ H0: p = ____ H0: p = ____

Alternate Hypothesis Ha: p ≠ ____ Ha: p < ____ Ha: p > ____

2. Conditions 1. Do we have a random sample [SRS]? 2. Are the observations independent [Pop > 10n à 𝝈𝒑 ok to calculate]? 3. Is the sampling distribution approx. Normal? [np & n(1-p) > 10 check]

3. Calculations

Test Statistic

𝑧 =𝑝 − 𝑝!𝑝!(1− 𝑝!)

𝑛

P-value

4. Conclusion Always refers to Ha in the end!!! Make sure it is in context to the problem!!!

When you do not know which way the alternate value could go, then we will perform a two-sided test with ≠.

Our test statistic could end up positive or negative, we just do not know.

One thing you CANNOT do is look at your data afterwards and see which direction the test statistic is at

compared to the hypothesized value and switch to a one-sided test…that’s statistical cheating!!!

One more thing to consider when performing a two-sided test: you now have two __________ to calculate the

area of to represent the ______________________. Since the normal distribution is ALWAYS

___________________________, then you simply have to ___________________ one tail’s area!!

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Example A coin that is balanced should come up heads half the time in the long run. The French naturalist Count Buffon (1707-1788) wanted to test this theory. He tossed a coin 4040 times. He got 2084 heads. Is this evidence that Buffon’s coin was not balanced!?!?! Perform a significance test at the 5% α-level.

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Why CI’s give more information [VIDEO #8] Consider the example from last video testing whether a coin was fair or not. We concluded that the coin was not fair because the p-value was less than the α-level. We know head’s comes up more often than tails as a result, but it does not give us any idea as to what the true proportion of flipping heads is for all instances.

Looking at the calculator output for a 95% CI and for a 5% hypothesis test reveals a special connection:

Notice: Is the hypothesized value of p = 0.5 being captured by the CI? ________ There will ALWAYS be this special connection between a CI and a two-sided hypothesis test. There is a connection between a CI and a one-sided test, but it’s not as clear cut as it is with a two-sided test. Example A friend of yours recently got a 73% on his last AP Stats test. He thinks the class average is about the same as his grade. You disagree with his assumption, but honestly have no idea if the class average is really higher or lower than his grade. Below are computer print-outs of a 95% CI and a hypothesis test at the 5% α-level based on a sample of 40 randomly selected test scores from the last test. What conclusion would you make regarding Ho = 0.73 vs. Ha ≠ 0.73?

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11.1 – Carrying out a Significance Test for µ [VIDEO #9]

In addition to performing the same “three C’s” we did for CI’s, you must do what?

__________________________________________________________________________________________







mention this information and STOP. It’s not worth calculating & concluding anything.

ii. IF data came from a sampling method not necessarily associated with high bias (like a convenience sample), then mention this and say “Proceed with caution!”

2. The observations of your sample must be independent.

a. Check the “10%” condition a.k.a. “Population > 10·n”. SHOW YOUR WORK! If it checks, then write “We may calculate the standard error.”

b. If NOT, then you need to use the Finite Population Correction formula we mentioned before but will not use in this class, so write “Standard error cannot be safely calculated.” STOP.

3. The sampling distribution of 𝑥 must be approximately normal. ***If you know σ somehow, then you can say and use the normal distribution (z-scores).

***If you do not know σ, then you can saw and use the t-distribution (t-scores). (More likely)

a. If the population is known to be Normal (REGARDLESS OF SAMPLE SIZE), then write “Since

the population is known to be normal, then we may assume the sampling distribution can

follow an (select one: approx. normal or t ) distribution.”

b. If NOT, then check for a sufficiently large enough sample size (> 30) for the CLT to apply

If so, write “Sampling distribution can use t-procedures due to the CLT and sample size.”

c. If sample size is less than 30: t procedures can be used except in the presence of outliers or

strong skewness (some skewness OK). Plot your data (stemplot or boxplot) to check for

Normality (roughly symmetric, single peak, no outliers). If the data checks out, then write

“Based on the data shown, we may assume a t-distribution with ___ df.”

d. If the data are STRONGLY skewed or if outliers are present, do not use t. STOP.

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“C”alculations: The “generic” formula that we used for proportion hypothesis tests: à turns into this for mean hypothesis tests… The “test statistic” is really a _____________ that says how “far away” our actual sample value is from our hypothesized population value. If it’s “far enough” away, then Ho is more than likely wrong! The p-value is the area under the curve “as extreme as or more extreme than” this t-score (test statistic). Remember how to find the area under the curve with lower and upper bounds??? _______________________ Draw a t-curve, find the approximate spot for the t-score, and then shade…WHICH WAY??? The shading depends on the ___________________________________________. If it uses ____, then shade to the ___________ of the t-score. If it uses _____, then shade to the __________ of the t-score. If it uses ____, then label both the positive AND negative t-scores and shade to the ____________ of those two t-scores. “C”onclusion: And if the p-value is ____ our pre-determined α-level, then we get to reject Ho in favor of Ha!!! Example Your favorite radio station claims to play an average of 50 minutes of music every hour. However, it seems that every time you turn to this station, there is a commercial playing. To investigate their claim, you randomly select 12 different hours during the next week and record what the radio station plays in each of the 12 hours. The sample mean, 𝑥, from your data is 47.9 minutes with a sample standard deviation, s, of 2.81 minutes. Assume the conditions have been met.

1. What must be true about our data to conclude it passed the normality check?

2. Calculate the test statistic.

3. Calculate the p-value.

4. What conclusion can you make at the 5% significance level? 1% α-level?

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11.1 – One-sided t-Test for µ [VIDEO #10] When performing a one-sided hypothesis test, keep in mind that the sign [negative or positive] that we get from calculating our test statistic IS NOW important. We don’t chop off the negative like we did with CI’s!!! Step #1: State hypotheses and define the parameter of interest. Step #2: “C”onditions Step #3: “C”alculations Step #4: “C”onclusion Example Caffeine, a chemical found in many popular beverages, is known for reducing fatigue. A college student wants to investigate his daily caffeine intake from beverages such as soft drinks, energy drinks, tea, and coffee… beverages this particular student consumes daily. If he becomes convinced that he consumes over 500 mg of caffeine per day then he will change his habits. Here is his daily caffeine consumption (in mg) for 14 days:

720 550 340 450 280 700 750 1050 380 1050 400 350 800 900

Perform a significance test at the α = 0.05 level.

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11.2 – Two-sided t-Test for µ [VIDEO #11] Step #1: State hypotheses and define the parameter of interest. Step #2: “C”onditions Step #3: “C”alculations Step #4: “C”onclusion The only BIG difference we will have to do for a two-sided vs. a one-sided test is to ____________ the p-value! Example In the children’s game Don’t’ Break the Ice, small plastic ice cubes are squeezed into a square frame. Each child takes turns tapping out a cube of “ice” with a plastic hammer, hoping that the remaining cubes don’t collapse. For the game to work correctly, the cubes must be big enough so that they hold each other in place in the plastic frame but not so big that they are too difficult to tap out. The machine that produces the plastic cubes is designed to make cubes that are 29.5mm wide, but the actual width varies a little. To ensure that the machine is working well, a supervisor inspects a random sample of 50 cubes every hour and measures their width. Use the information from the print-out to test this claim. Also, verify your conclusion with the 95% CI.

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11.2 – Matched Pairs t-Test for µ [VIDEO #12] Police trainees were seated in a darkened room facing a projector screen. Ten different license planes were projected on the screen, one at a time, for 5 seconds each, separated by 15-second intervals.

After the last 15-second interval, the lights were turned on and the police trainees were asked to write down as many of the 10 license plate numbers as possible, in any order at all.

A random sample of 15 trainees who took this test was then given a week-long memory training course. They were then retested.

Test, at the 5% significance level, whether the memory course improved the ability of the trainees to correctly identify license plates.

Officer Plates correctly Identified before

training

Plates correctly Identified after

training 1 6 6 2 5 8 3 6 6 4 5 7 5 7 9 6 5 8 7 4 9 8 6 6 9 7 7

10 8 5 11 4 9 12 5 8 13 4 6 14 6 8 15 7 6

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Upon completion of Chapters 10 - 12, you should be able to: ! Determine whether a specific p-value is significant at the .05 or .01 level, both or neither level. ! Find a p-value given a hypothesis and a test statistic. ! Carry out a hypothesis test from a CI. Remember, the alternative hypothesis must be two sided and the α

level must be complementary with the CI level. See your notes if you forget how to state the conclusion when carrying out a test from a CI.

! Explain the differences between Type I Error, Type II Error, and Power. Know the meaning of each and that Type I Error is just α, and that Type II Error and Power are complementary. Also know that you must have a fixed α in advance and that values far from Ho are easier to detect (higher power) than those close to Ho.

! Carry out a complete hypothesis test. Remember EVERY component of a hypothesis test. ! Set up the null and alternative hypothesis and define µ for a hypothesis test. ! Check conditions. ! Calculate the test statistic and p-value. Remember – Double the p-value for 2-sided tests. ! Write a concluding statement for a hypothesis test in the context of the problem. ! Describe Type I and Type II error in regards to the hypothesis test you carry out. ! Reduce the probability of Type II Error and increase the power of a test. What can be done to obtain high

power (low type II error) for any hypothesis test? ! Make sure you know your formulas and don’t get the components of them mixed up. You should know

them off by heart from doing HW problems. ! How to verify that the population is normal with either a stem-plot or a histogram, be sure to note the

absence of outliers and strong skewness ! How to increase the power of any test ! Tell the difference between a one sample t-test and a matched pairs t-test ! Calculate the standard error for both a one sample proportion. ! Interpret the p-value of a hypothesis test

chapter 10 - 12 – significance/hypothesis testing · 10.2 – 10.4 [watch chapter 11 videos 1 –...

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