Practice exam #3

        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 (m) for sick days per year taken at the company is 4 with a population standard deviation (s) 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 H₀, H₁, m₁, and m₂, write out the research hypothesis and null hypothesis for this example. Assume that m₁ refers to the population mean for vitamin takers and m₂ 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 ________.

You want to see if New Paltz students who smoke take tend to drink more beer per week than New Paltz non-smokers. 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?

12. In addition to all these constructs, be sure to know the distinction between Type I and Type II Error.

ANSWERS:

1. H₁: m₁ < m₂

H₀: m₁ >= m₂

2. Z_critical = -2.33

s_M = Sq. rt: s²/N = Sq. rt: 1²/10 = .32

M_critical = -2.33(.32) + 4 = 3.26

Z = (3.26-3)/.32 = .82

Power = .79

3. Beta = 1-power = .21

4. Cohen's d = (m₁ - m₂)/s = (3-4)/1 = -1 (or just 1) ... it’s large

5. SS = Sum of(X-M)² = 10

s² = SS/(N-1) or SS/df = 10/3 = 3.33

s = Sq rt: s^{2 =} 1.82

s²_M = 3.33/4 = .83

s_M = .91

t = (M - m)/s_M = (10-6)/.91 = 4.40

6. t_critical = 2.35 (df = 3, one-tailed, p < .05)

7. reject Ho

8. t = (mean difference score)/s_M = -1/.58 = -1.72

... here's how to get s_M:

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

s² = SS/(N-1) = 10/5 = 2

s_M = Sq Rt:(s²/N) = Sq Rt:(2/6) = .58

9. t_critical = -2.01 (df = 5, one-tailed, p < .05)

10. Fail to reject

11. Cohen's d = (mean difference score)/s = -1/1.41 = -.71 (or just .71)