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Hypothesis Testing

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Hypothesis Testing

Comparisons Among Two Samples

- By far a more common statistic to use than any other covered so far
- If we have access to an entire population to calculate its mean (μ), why do we need to get a sample to infer its characteristics? We can just measure them directly
- 99% of the time we don’t have this kind of access

- Also, most experiment employ two groups – a treatment group and a control group
- The Treatment Group – gets the IV
- The Control Group – identical to the Tx Group, minus the IV
- Since Control = Tx + IV, if our Tx Group has a different mean than our Control Group, we can attribute this to the IV

- Ex. If we want to know what effect fire has on water, we have two groups, one that has water over an open flame (the Tx Group), and one that has water unheated (the Control Group). If all other factors that could influence the heat of water are kept constant between the two groups (i.e. the water in the two groups has the same salinity, the air pressure is the same, etc.), if the water in the heated condition is hotter, we can conclude that the difference is due to the fire, and not the other factors that we kept constant

- Assumptions of the Two-Samples T-Test:
- 1. Normally-distributed data
- Like all other t-tests, and according to the Central Limit Theorem, as long as sample sizes are large, you can ignore this requirement

- 1. Normally-distributed data

- 2. Homogenous Variances (s2)
- As long as neither of your variances (s2) are more than four times the other (you have two variances, one for each sample) and your samples have similar sample sizes (n), you’re OK
- If both: one of your variances is four or more times the other and you have unequal sample sizes, then you have to use another procedure

- 3. Independent Samples
- As stated previously, your groups have to be independent – i.e. a subject can only be in one group and the group cannot be yolked
- Yolked Groups = you assign subjects to groups that look similar on a variable or variables that you’re interested in
- Ex. You’re interested in sociability ratings, so since subject 1 in Group A has a sociability rating of 45, the next subject that has a similar rating you assign to Group B
- Here, membership in Group B is dependent on the subjects in Group A, i.e. the groups are not independent

- Yolked Groups = you assign subjects to groups that look similar on a variable or variables that you’re interested in

- As stated previously, your groups have to be independent – i.e. a subject can only be in one group and the group cannot be yolked

- How to compute a Two-Samples T-Test?
- Instead of subtracting μ from , you subtract from
- +t = >
- -t = <

- Also, instead of using s2, we use
- n for grp 1 variance for grp 1 n for grp. 2 variance for grp 2
- This is what is called our Pooled Variance

- This formula averages the variances from our two samples, however…
- First it multiplies the variance by the sample size [s2 x (n-1)], which gives more importance to variances from larger samples

- This is an example of what is called a weighted average
- Weighted Average = average where our values to be average are multiplied by a factor that we think is important (in this case, n is this factor)

- Therefore, our formula for our Two-Samples T-Test is:

- Since we’re using two samples, our
df =

- Also, the form of our hypothesis changes:
- For a One-Tailed Test:
- H0 = (μ1 > μ2) (or visa-versa)
- H1 = (μ1≤ μ2) (or visa-versa)

- For a Two-Tailed Test:
- H0 = (μ1 = μ2)
- H1 = (μ1 ≠μ2)

- For a One-Tailed Test:

- We would apply the Two-Samples T-Test in the same way as previous tests:
- 1. Identify H0 and H1
- 2. Calculate df and identify the critical t
- 3. Determine whether to use one- or two-tailed test, determine what value of α to use (usually .05), and identify the rejection region(s) that the critical t is the boundary of

- 4. Calculate the variances for both samples (s1 and s2), and use them to calculate the pooled variance
- 5. Calculate the mean of both samples
- 6. Utilizing this information, calculate t
- 7. Compare your value of t to your critical value and rejection region to determine whether or not to reject H0

- Confidence Intervals:
- The formula is the same as the one-sample t-test, once again all that is different is that we use two sample means instead of a sample mean and a population mean, and a pooled variance instead of a regular, sample variance
- CI = ± Critical t (two-tailed at p=.05) x pooled standard deviation

- The formula is the same as the one-sample t-test, once again all that is different is that we use two sample means instead of a sample mean and a population mean, and a pooled variance instead of a regular, sample variance

- Example:
- Much has been made of the concept of experimenter bias, which refers to the fact that for even the most conscientious experimenters there seems to be a tendency for the data to come out in the desired direction. Suppose we use students as experimenters. All the experimenters are told that subjects will be given caffeine before the experiment, but half the experimenters are told that we expect caffeine to lead to good performance, and half are told that we expect it to lead to poor performance. The dependent variable is the number of simple arithmetic problems the subject can solve in 2 minutes. The obtained data are as follows:

- What are the Ho and H1?
- What is the df and critical t?
- What are your alpha, type of test (one- vs. two-tailed), and rejection region(s)?
- What is your t?
- Will you reject or fail to reject the null hypothesis?

- Assuming a one-tailed test, H0 = (μ1 > μ2); H1 = (μ1≤ μ2), where μ1 = “Expect Good Performance”
- df = 15; critical t(15) = 1.753
- t = .587
- Fail to reject the null hypothesis
- Our data do not support the theory that experimenter bias influences data.