Welcome to MM570 Psychological Statistics Unit 5 Introduction to Hypothesis Testing Dr. Ami M. Gates

Normal Probability Distribution Curve Sample Curve μ (mean) = 500 σ (st. dev.) = 100 How many Standard Deviations away from the mean is: SAT Scores Standard Deviations

Z Scores A z-score (or z value) is the number of standard deviations above or below the mean  Here, our z-value “Z” for any datapoint “X” is that datapoint X minus the mean of the dataset, divided by SD the standard deviation of the dataset.

For Example Suppose our dataset has mean of 500 and st. dev of 100 μ = 500 σ = 100 How many Standard Deviations away from the mean is: X = 300? ( )/100 = -200/100 = -2

Example 2: Suppose we have a mean of 500 and st dev of 100 μ = 500 σ = 100 How many Standard Deviations away from the mean is: X = 650? ( )/100 = 150/100 =

The Normal Distribution and the Empirical Rule 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%

What is Hypothesis Testing Most research starts with an “idea”. This “idea” is then formalized into a “null and alternative (research) hypothesis”. Once we have a hypothesis, we can then collect samples and test our theory. We can use tests like the z-test or t-test to determine if we should accept or reject our null hypothesis.

Here is an Example of how a Hypothesis Test starts: Suppose a researcher thinks that not eating red meat will significantly lower cholesterol. This thought is the idea that is used to build the hypothesis test. A hypothesis test always has a null hypothesis (Ho:) and an alternative (research) hypothesis (Ha:) Here, our research idea is: Not eating red meat will significantly reduce cholesterol levels in men over 45 in the USA.

Next, this idea becomes a formal null and alternative (research) Hypothesis Here, our research idea is: Not eating red meat will significantly reduce cholesterol levels in men over 45 in the USA. But! We need to make this specific and put it in terms of null and alternative hypothesis. Our first goal is to decide how we plan to run or test our idea or experiment. In this case, 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, I do not group 2 eat any meat. After 6 months, I collect everyone’s cholesterol. Group 1 (red meat OK) Group 2 (CANNOT eat red meat)

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)

Here, we want to compare our groups! So far, we have collected a sample of 30 men from our population of all men over 45 in the USA. Then, of these 30 men, we created two groups of 15 men each Group 1 (red meat OK) Group 2 (CANNOT eat red meat) Next, we ran our experiment for 6 months. During our 6 months, group 1 can eat normally but group 2 cannot eat red meat. After the 6 months ends, we measured everyone’s cholesterol. Next, we can calculate the mean cholesterol for each group.

Using SPSS to get the mean for each group So, after 6 months, we have: Group 1 (meatOK) mean = Group 2 (no meat) mean =

Defining the null and alternative (research) 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. The null hypothesis (written as Ho: ) is always the status quo or what is currently accepted to be true. The alternative or research hypothesis (Ha: ) is what YOU are trying to prove. The Ho and Ha are ALWAYS opposite! In our case: 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)

Now we have our exact hypothesis test with Ho and Ha In our case: 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) Notice that Ho and Ha are opposite Also notice that Ho uses an “=“ sign. The null hypothesis ALWAYS contains an “=“ sign and represents what is commonly believed. Notice that in this case the alternative (research) hypothesis Ha contains a ≠ sign. This means that our hypothesis test is two-tailed! An alternative or research hypothesis can have “>”, “<“, OR “≠” If the alternative hypothesis has “>” or “<“ it is called one-tailed.

Let’s review! We start with a research “idea” We formalize and clarify this idea and use it to create our null (Ho) and alternative (Ha) hypothesis The null and alternative are always opposite The null always has an “=“ and represents the status quo or what is commonly believed. Example: Ho: mean cholesterol of Group 1 (meat OK) = mean cholesterol of Group 2 (no meat) The alternative (research) hypothesis can have the following signs: ≠ “two-tailed test” > “one-tailed test” < “one-tailed test” For example: Ha: mean cholesterol of Group 1 (meat OK) ≠ mean cholesterol of Group 2 (no meat) Our test is a two-tailed test

Let’s get back to our research question about cholesterol In our case: 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) Remember, we waited 6 months and then calculate the mean cholesterol for each group. Recall that our means from SPSS were: Group 1(meat OK) has mean cholesterol = Group 2(No Meat) has mean cholesterol = Do we “REJECT THE NULL HYPOTHESIS in favor of the alternative”?? In other words, can we conclude that our research idea is significant? Should people stop eating meat to change their cholesterol??

To determine if we should accept or reject the null hypothesis, we “run a test” Remember that all hypothesis testing uses samples! In our case, we have two sample groups of men that are of size 40 each. But!! Our population of men is over a billion. Samples – no matter how large – are always MUCH smaller than the population that they come from. So, even when we look at the means from our samples: Group 1(meat OK) has mean cholesterol = Group 2(No Meat) has mean cholesterol = We still need to run a statistical test to see if these sample means are significantly different and if we can reject Ho in favor of our research hypothesis.

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. Recall – we have: Group 1(meat OK) has mean cholesterol = Group 2(No Meat) has mean cholesterol = If they are significantly different, we will reject the null. If they are not significantly different, we will NOT reject the null. Remember, the null was: Ho: mean cholesterol of Group 1 (meat OK) = mean cholesterol of Group 2 (no meat)

We can use SPSS to run our t-test (Yay!) Here, our t = Also, our Sig (2 –tailed) =.000 What do these values mean?

Using Alpha and our t test We 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 For a 2-tailed test We have.05/2 or 2.5% on each tail that is our rejection region

But how do we determine the rejection values or critical values?? We have two choices The first is to use a table like this: But!! We are using SPSS and SPSS will tell us the p- value. The p-value is the “result” of whether you can reject the null of not. SPSS calculates the p-value using these table values. If the p-value given by SPSS is <.05 (our alpha), then we REJECT THE NULL! Otherwise, we DO NOT reject the null

In our example, the p-value = 0 < alpha of.05 so we can reject the null and conclude that meat DOES affect cholesterol! In SPSS, the p-value is called “Sig (2-tailed)” In this case the Sig is.000 Because.000 <.05 we reject the null and conclude that there is sufficient evidence to support our research hypothesis. Cholesterol is affect by meat consumption!

How does SPSS get the p-value or Sig? SPSS has in its memory the table of values for the rejection value of the t-test. If the t-test result falls in the rejection area (in the gray area as seen below) then the p-value (called Sig) will be less than.05 (our alpha value). If this happens, we can reject the null. If the p-value (called Sig) is >.05 We cannot reject the null And that’s it!!

In our example, we reject the null! Notice that our t test result is 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 : and Notice that > and so it IS in the rejection region. Our Sig or p-value of.000 tells us this because.000 <.05

Visually, here is what SPSS is doing:

The Good News: We can use the SPSS p-value (called Sig) to determine if we reject or do not reject the null Ho Step 1: Run the test Step 2: Determine if you have equal variance – we will talk about this in Unit 6. Step 3: Find the Sig value. If the Sig value <.05 you can reject the null. You do not need to use the table ;)

Conclusions Our t-test showed us that we can reject the Null hypothesis and can choose our research hypothesis. Therefore, we can say that we have showed that no eating meat does affect cholesterol levels. Finally, we can look at both means to see that the “no meat” group had lower cholesterol