Question

Independent-Samples t-Test

In a research project, researchers track the health and cognitive functions of the elderly in the community. To examine any possible gender differences in their sample, they want to see if the females and the males differ significantly on the education level (number of years of formal schooling). The researchers are not predicting any direction in the possible gender differences so the hypotheses should be non-directional. They would like to run a two-tailed test with α = .10.

Note: In the statistical notations below, 1 = male, 2 = female, so sample 1 has the males while sample 2 has the females.

Subject ID	Gender	Education		Subject ID	Gender	Education
1	1	10		11	2	12
2	1	12		12	2	9
3	1	12		13	2	13
4	1	16		14	2	13
5	1	15		15	2	10
6	1	14		16	2	14
7	1	16		17	2	15
8	1	13		18	2	11
9	1	16		19	2	16
10	1	16		20	2	15

a. What is the dependent (outcome) variable? What is the independent (grouping) variable? (2 points total: 1 for each variable)

b. Create the null and alternative hypotheses (non-directional) for this study, using both words and symbol notation (2 points total: 1 for each hypothesis, with .5 for written and .5 for symbol notation)

c. Calculate M1 and M2 (2 points total: 1 point per sample mean, .5 if process is accurate but the result is calculated incorrectly)

d. Calculate df1 , df2, and dftotal (1 point total: deduct .5 for each error)

e. Calculate estimated variance for population 1 (s12) and estimated variance for population 2 (s22) (2 points total: 1 point for each variance, .5 if the process is correct but the answer is wrong)

f. Calculate the pooled variance (Spooled2) from the two population variances (from question e above) (1 point total: .5 if the process is correct but the answer is wrong)

g. Use the pooled variance (from question f above) to calculate the variance for sampling distribution 1 (SM12) and the variance for sampling distribution 2 (SM22) (2 points total: 1 for each variance, .5 if the process is correct but the result is calculated incorrectly)

Hint: Sampling distribution is derived from the original population and it consists of means of all possible samples drawn from the original population.

h. Calculate standard deviation (Sdiffmean)of the comparison distribution (1 point total: .5 if the process is correct but the answer is wrong)

Hint: This comparison distribution consists of differences between all possible sample means drawn from the two sampling distributions. Its standard deviation is the denominator of the t statistic formula.

i. Calculate the t statistic (1 point total: .5 if the process is correct but the answer is wrong)

j. For the two-tailed test, find the critical t values for this hypothesis test based on the total degree of freedom (from question d above), and the preset alpha level. (1 point total)

k. Compare the calculated t statistic with the critical t value by stating which is more “extreme”, and then draw a conclusion about the hypothesis test by stating clearly “reject” or “fail to reject” the null hypothesis. (2 points total: 1 for comparison, 1 for decision)

l. Calculate the pooled standard deviation for the populations (use the pooled variance calculated in question f); and then calculate the standardized effect size of this test. (2 points total: 1 for pooled standard deviation, 1 for effect size. Deduct .5 if a result is wrong but the process is correct.)

m. Based on the hypothesis test, is there a significant difference in education level between the males and the females in the sample? If so, in what direction is the gender difference? (1 point total: .5 for each answer)

Answer 1

a. dependent variable = education

independent variable = gender

b.The null and alternative hypothesis

H₀ :

Ha :

c) M1= (10+12+........+16)/10=14

M2 = (12+9+.......+15)/10 =12.8

d) df1 = 9

df2 = 9

dftotal = 18

e)s₁² ={ (10-14)² +.......(16-14)²}/9 = 4.67

s₂² = {(12-12.8)² +......(15-12.8)²} /9 = 5.29

f) s_pooled ²= (9*s₁² +9*s₂²)/ (10+10-2) = 4.9778

g) SM₁ = s_pooled / sqrt(10) =0.7055

SM2  = s_pooled / sqrt(10) =0.7055

h) s = sqrt ( 4.9778) *sqrt ( 1/10+ 1/10) = 0.9978

i) t= (M₁ - M₂) / s = 1.20

j) for alpha = 0.10 and df = 18

t_{critical = 1.73}

k) t_{critical is more extreme}

fail to reject null hypothesis