Study Guide for Exam #3

For the computational components related to power, the following examples are provided:

You want to know whether employees of a company who take vitamins every day take, on average, less sick days than employees at the company in general. The population mean (μ) for sick days per year taken at the company is 4 with a population standard deviation (σ) of 1. Suppose that the mean number of sick days for vitamin-takers, in actuality, is 3. In order to test your hypothesis, you found the number of sick days taken by 10 vitamin-takers and used an alpha level of p < .01.

1. In terms of H_{0}, H_{1}, μ_{1}, and μ_{2}, write out the research hypothesis and null hypothesis for this example. Assume that μ_{1} refers to the population mean for vitamin takers and μ_{2} refers to the population mean for employees in general.

2. What would your power equal in this example?

3. What would Beta equal in this example?

4. What is the effect size (Cohen’s d) for this example?

According to Cohen’s effect size convention, this effect size is ________.

MAKE SURE TO FULLY UNDERSTAND TYPE-I and TYPE-II error for this exam. That content is introduced in Chapter 7 but it comes back for Chapter 9 on statistical power.

- · You want to see if New Paltz students who smoke take tend to drink more beer per week than New Paltz students in general. You know that the population mean (m) for New Paltz students is 6. However, you have no idea what the population standard deviation (s) is. You randomly ask 4 smokers how many beers they drink per week. Assume an alpha level of p < .05. Here are your results:

X

8

12

11

9

5. What is t in this example?

6. What is t_{critical} in this example?

7. What do you conclude about the null hypothesis. EXPLAIN.

- · You want to test the Popeye hypothesis: You are pretty sure that eating spinach makes people stronger. In order to test this hypothesis, you count the number of pushups that 6 people can do. Then you make them all eat spinach. Then you count how many pushups they can do after they eat spinach. Assume an alpha level of p < .05. Here are the number of pushups they did before and after the spinach:

Pre-spinach Post-spinach

4 5

3 3

7 9

2 3

5 8

6 5

8. What is t in this example?

9. What is t_{critical} in this example?

10. What do you conclude about the null hypothesis. EXPLAIN.

11. What is the effect size (Cohen’s d) for this example?

- · You want to know whether there is some difference in the amount of sporting events attended for different groups of people: (a) physical education majors, and (b) business majors. In order to test this hypothesis, you randomly select members of each group and find out how many sporting events they each have attended in the past year. You assume an alpha level of p < .01. Here are the scores:
- · number of sporting events attended in past year:

(Physical education majors)

X_{2}

8

10

12

(Business majors)

X_{3}

0

0

3

- · You want to know whether physical education majors attend more sporting events per year than business majors. Assume an alpha level of p < .01.

12. Calculate t.

13. What is t_{critical} in this example?

14. What is your decision concerning the null hypothesis? EXPLAIN in terms of whether the means for the two groups you are comparing are significantly different from one another.

15. What is the effect size in terms of Cohen’s d for this example?

Cohen’s d = ________

16. In terms of Cohen’s conventions for effect size, this effect size is ______.

FORMULAS/ANSWERS:

1. H_{1}: μ_{1} < μ_{2}

H_{0}: μ_{1} >= μ_{2}

2. Z_{critical} = -2.33

σ^{2}_{M} = σ^{2}/N = 1^{2}/10 = .1; σ_{M} = Square Root of (.1) = .32

M_{critical} = -2.33(.32) + 4 = 3.25

Z_{upper} = (3.25-3)/.32 = .78

Power = 78.23%

3. Beta = 100%-power = 21.77%

4. Cohen’s d = | (M – μ_{2})/ σ | = (3-4)/1 = 1 … it’s large

5. SS = ∑(X-M)^{2} = 10

s^{2} = SS/(N-1) or SS/df = 10/3 = 3.33

s = Square Root of s^{2 = }1.82

s^{2}_{M} = 3.33/4 = .83

s_{M} = .91

t = (M -µ_{M2})/s_{M2} = (10-6)/.91 = 4.40

6. t_{critical} = 2.35 (df = 3, one-tailed, p < .05)

7. reject Ho

_{ }

8. t = (đ)/s_{M} = -1/.57 = -1.75

… here’s how to get s_{M2}:

SS (of the difference scores … subtract each difference score from the mean of the difference score; then square it; then sum these squared numbers) =

SS = 10

s^{2} = SS/(N-1) = 10/5 = 2

s^{2}_{M} = s^{2}/N = 2/6 = .33

s_{M2 = Square root of = .57
}

9. t_{critical} = -2.01 (df = 5, one-tailed, p < .05)

10. Fail to reject

11. Cohen’s d = | (đ)/s |= -1/1.41 = -.71 (or just .71)

12. t_{obt} = (10-1)/1.53 = 5.88

13. t_{critical} = 3.75

14. Reject Ho … the mean number of sporting events attended by phys. ed. majors is significantly greater than the mean number attended by business majors.

15. Cohen’s d = | (10-1)/1.87 |= 4.81

16. In terms of Cohen’s conventions for effect size, this effect size is gigantic.