Welcome to MM570Welcome to MM570Psychological StatisticsPsychological Statistics

Welcome to MM570Welcome to MM570Psychological StatisticsPsychological Statistics

Unit 5 SeminarDr. Rhoda Deon

The Normal DistributionThe Normal Distribution

• Normal curve and approximate percentage of scores between the mean and 1, 2, and 3 standard deviations from the mean* Other texts may refer to the Empirical Rule or

68% - 95% - 97.5%

Sample and PopulationSample and Population

• Population• Sample• Methods of sampling• Random selection• Haphazard selection

ProbabilityProbability

• Range of probabilities• Proportion: from 0 to 1• Percentages: from 0% to 100%

• Probabilities as symbols• P • p < .05

• Probability and the normal distribution• Normal distribution as a probability distribution

What is Hypothesis Testing?What is Hypothesis Testing?

• Any hypothesis is a “guess” about some condition, Behavior Therapy is more effective than Psychotherapy or the average time that Kaplan students in MM570 spend studying is 12 hours per week.

• How do we know if our guess is correct? We “test” it using data.

• Will our data prove the guess is correct or incorrect? No, it could be wrong.

• We use statistics to determine the likelihood that our data either supports or does not support the guess.

The Concept of Hypothesis TestingThe Concept of Hypothesis Testing

• You and I are going to play a game flipping my coin. If it lands heads I will give you a dollar. If it lands tails you give me a dollar.

• Now let’s assume you agree to play my game. You are making an assumption (guess) about my coin, that it is fair!

• The assumption that it is fair is what we will test. It is called the Null Hypothesis.

• We could state this as the probability of the coin landing heads is .50 and landing tails is .50

The Null and Alternative HypothesesThe Null and Alternative Hypotheses

• We restate our “guess” so we can test it. • There are many ways our guess can be wrong but

only one way it can be correct

• We write the Null Hypothesis as a statement• Ho: p(heads) = .50

• The Alternative would be that the coin is not fair• Ha: p(heads) ≠ .50

• Notice that these two statements include all possible outcomes of our “test.”

Collecting DataCollecting Data

• We need to collect data to determine which hypothesis, is the coin fair or not, to accept• I flip the coin and it is heads, I give you $1• I flip the coin and it is heads, I give you $1• I flip the coin and it is tails, you give me $1• I flip the coin and it is heads, I give you $1• I flip the coin and it is tails, you give me $1

• At this point do the data seem to indicate that the coin is fair? Yes I think we would agree it does.

Making a Decision with DataMaking a Decision with Data

• We continue the game• I flip and it is heads, I give you a $1• I flip and it is tails, you give me a $1• I flip and it is tails, you give me a $1• I flip and it is tails, you give me a $1• I flip and it is tails, you give me a $1

• Does the coin still seem to be fair? It is not as clear as it was but maybe it is fair.• We flip 10 more times and you lose all of them

• Now at this point you reject the claim that the coin is fair and call me a cheat. But, could you be wrong and the coin really is fair? The answer is YES!!

What Have We Done?What Have We Done?

• Started with an assumption• Created a null and alternative hypothesis• Collected data related to our question• Based on the data we either rejected or accepted

the null hypothesis (this is the one we test and if we reject it, we are accepting the alternative)

• Made a decision about the original “guess” with appropriate consideration of the fact that you could be wrong!

An Example of the Test of HypothesisAn Example of the Test of Hypothesis

• Suppose a researcher thinks that not eating red meat will significantly lower cholesterol. This assumes that eating red meat will not lower cholesterol

• This leads us to an experiment comparing two groups, those not eating red meat and those who do eat red meat.

• Now, suppose I take a sample of 30 men (over age 45 from the USA) and separate them in to two groups, each with 15 men. Then, for 6 months, group 2 does not eat any red meat. After 6 months, I collect everyone’s cholesterol.• Group 1 (red meat OK)• Group 2 (CANNOT eat red meat)

The Null and Alternative Hypothesis:

Ho and Ha

The Null and Alternative Hypothesis:

Ho and Ha• Remember, our research idea is that not eating meat will significantly reduce cholesterol. But, we need to create our Ho and Ha.

• Remember that the Ho and Ha are ALWAYS opposite!

• In our study we would have:• Ho: mean cholesterol of Group 1 (meat OK)

= mean cholesterol of Group 2 (no meat)• Ha: mean cholesterol of Group 1 (meat OK)

≠ mean cholesterol of Group 2 (no meat)

One-Tailed and Two-Tailed TestsOne-Tailed and Two-Tailed Tests

• Notice that Ho uses an “=“ sign. The null hypothesis ALWAYS contains an “=“ sign.

• In our study the alternative or research hypothesis Ha contains a “≠” sign. This means that our hypothesis test is two-tailed!

• An alternative or research hypothesis can have “>”, “<“, or “≠” sign. If the alternative hypothesis has “>” or “<“ it is called one-tailed.

Here is what our “data in SPSS” might look like after the 6 months are over and I have collected everyone’s cholesterol values.

*Notice that some people are in Group 1 (meat OK) and some are in Group 2 (no meat)

What we want to do is to compare the mean cholesterol levels for each group.

Here is what our “data in SPSS” might look like after the 6 months are over and I have collected everyone’s cholesterol values.

*Notice that some people are in Group 1 (meat OK) and some are in Group 2 (no meat)

What we want to do is to compare the mean cholesterol levels for each group.

Using SPSS to get the mean for each group

Using SPSS to get the mean for each group

So, after 6 months, we have:

Group 1 (meat OK) mean = 255.40

Group 2 (no meat) mean = 143.47

Returning to our research question about cholesterol

Returning to our research question about cholesterol

Group 1(meat OK) has mean cholesterol = 255.40

Group 2(No Meat) has mean cholesterol = 143.47

Now we can clearly see that 255.40 is not equal to 143.47! But they are based on only a sample of 30 men over 45. There are over millions of men over the age 45! Samples are always much smaller than the population from which they are drawn.

Is the difference we see in our small sample something that happened by chance (the Null is actually true) or is the difference real (the Alternative is actually true)?

This is what our “test” can help us determine.

What “tests” can we run to see if we should reject or accept the null?What “tests” can we run to see if we should reject or accept the null?• In this case, we have two samples and we are comparing the means

of these two samples.

• We do not really have any other information – like a population variance – so it is best to use the “t-test” to compare two sample means.

• Group 1(meat OK) has mean cholesterol = 255.40• Group 2(No Meat) has mean cholesterol = 143.47

• If they are significantly different, we will reject the null that the means are equal. If they are not significantly different, we will NOT reject the null and by extension accept the alternative that they are not equal.

We can use SPSS to run our t-test(Yay!)We can use SPSS to run our t-test(Yay!)

Here, our t = 8.148

Also, our Sig (2 –tailed) = .000

What do these values mean?

Using Alpha and our t testUsing Alpha and our t testWe use “alpha” = .05 to determine the rejection region for our t-test result.

We “reject the null” if our t test value is in our rejection region (sometimes called the critical region)

Our “rejection region” for the t-test is a curve that is very similar to the normal bell curve.

When alpha = .05 and a2-tailed test we have .05/2 or 2.5% in each tail that is our rejection region

Determining the rejection values or critical values?Determining the rejection values or critical values?We have two choices The first is to use a table like this:

Using SPSS Instead of the Table Using SPSS Instead of the Table

• SPSS will tell us the p-value (Sig) which is the probability that the difference is not just a chance occurrence.

• The reported significance is tells us whether to reject the null or not. SPSS calculates the p-value using these table values.

• If the p-value given by SPSS is less than (<) .05 (our alpha), then we REJECT THE NULL!

• If the Sig > .05 we DO NOT reject the null, the results are likely (more than 5 times in 100) if there really is no difference.

Interpreting the SPSS OutputInterpreting the SPSS Output

Because the sig or p-value = .000 is well less than our alpha of .05 we can reject the null and conclude that there is sufficient evidence to support our research hypothesis that not eating red meat does lower cholesterol.

Have we “proven” that not eating red meat lowers cholesterol? No we have not. The probability is very small but it is never 0, and therefore whatever we decide there is some small proability that we could be wrong!

SPSS and the Critical ValuesSPSS and the Critical Values

Notice that our t test result is 8.148, equal variances assumed.

If you use the table and not SPSS with alpha = .05 and df = 28, you will find that for a two-tailed test, the critical values or rejection values are : 2.048 and -2.048

We see that 8.148 > 2.048 and so it IS in the rejection region.

It is same conclusion we got using the Sig or p-value of .000.

Visually, here is what SPSS is doing:Visually, here is what SPSS is doing:

Conclusions from the Test of Hypothesis

Conclusions from the Test of Hypothesis

• Our t-test showed us that we can reject the Null hypothesis and can accept our research hypothesis.

• Therefore, we can say that the evidence supports the statement that not eating meat does lower cholesterol levels.

Notice that we did two things here. First we made a decision, based on the output from SPSS, about the Null Hypothesis. Second, we made a statement about our research question in the language of the question based on our decision. The end is NOT just accepting or rejecting the Null!!!!

QUESTIONS??QUESTIONS??