Question

A research project has been tracking the health and cognitive functions of the elderly population in Arizona. The table below shows the memory test scores from 16 elderly residents, tested first when they were 65 years old and again when they were 75 years old. The researcher wants to know if there is a significant decline in memory functions from age 65 to age 75 based on this sample. In other words, it is hypothesized that the memory score at age 75 is significantly lower than the memory score at age 65. So the null and alternative hypotheses should be directional. The alpha level was set at α = .05 for a one-tailed hypothesis test.

	Memory score
Subject	Age 65	Age 75
1	62	60
2	95	88
3	55	56
4	90	89
5	98	90
6	73	75
7	73	70
8	71	75
9	82	80
10	66	62

a. Identify the dependent variable (this is the outcome measure) and the independent variable (this is what differentiates the two groups of data points being compared). (1 point total: .5 for DV, .5 for IV)

b. Explain why a paired-samples t test is appropriate for answering this research question. (1 point)

c. What would be the null and alternative hypotheses in both words and symbol notations? (2 points total: 1 for each hypothesis, .5 for written and .5 for notation)

d. Calculate the difference score by subtracting each “Age 65” score from the associated “Age 75” score for each subject. Fill in the column in the table below for “difference score.” (1 point total: deduct .5 for each error up to 1 point.)

Hint: The difference score is calculated as (age 75 minus age 65), so a negative number indicates a decline in memory performance, which is the researcher’s hypothesis.

Subject	Difference score (Age 75 – Age 65)
1
2
3
4
5
6
7
8
9
10

e. Calculate the mean from the sample of difference scores (1 point total: .5 if process is correct but answer is wrong)

f. Estimate the standard deviation of the population of difference scores (1 point total: .5 if process is correct but answer is wrong)

g. Calculate the standard error (standard deviation of the sampling distribution) (1 point total: .5 if process is correct but answer is wrong)

h. Calculate the t statistic for the sample of difference scores (1 point total: .5 if process is correct but answer is wrong)

i. Figure out the degree of freedom, and then determine the critical t value(s) based on the type of test and the preset alpha level. (1 point total: .5 for df, .5 for critical t value)

j. Compare the t statistic with the critical t value. Is the calculated t statistic more extreme or less extreme than the critical t value? Then make a decision about the hypothesis test, stating explicitly “reject” or “fail to reject” accordingly. (2 points total: 1 for each answer)

k. Interpret the result in 1-2 sentences to answer the research question (you may use the wording from the hypothesis or explain it in your own words) (1 point)

l. Calculate the standardized effect size of this hypothesis test (1 point: .5 if process is correct but answer is wrong)

Answer 1

Ans:

a)dependent variable is memory score.

independent variable is age.

b)As,samples are dependent,same individual is measured at different times.

c)d=Age 75-Age 65

H₀: The memory score at age 75 is equal to or greater than the memory score at age 65

Ha:The memory score at age 75 is lower than the memory score at age 65

d)

Subject	Age 65	Age 75	d
1	62	60	-2
2	95	88	-7
3	55	56	1
4	90	89	-1
5	98	90	-8
6	73	75	2
7	73	70	-3
8	71	75	4
9	82	80	-2
10	66	62	-4
		d-bar	-2
		sd	3.771

e)sample mean for difference=-2

f)sample standard deviation for d=3.771

g)

t=(-2-0)/(3.771/sqrt(10))

t=-1.677

h)df=10-1=9

i)critical t value=-1.833

Reject Ho,if t<-1.833

j)As,test statistic,t does not fall in rejection region,we fail to reject the null hypothesis.

k)There is not sufficient evidence to conclude that the memory score at age 75 is significantly lower than the memory score at age 65.

l)effect size=2/3.771=0.53