A geriatric 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 based on this sample. The alpha level was set at α = .05 for the hypothesis test.
Memory score 

Subject 
Age 65 
Age 75 
1 
62 
60 
2 
95 
62 
3 
55 
46 
4 
97 
89 
5 
98 
88 
6 
63 
80 
7 
73 
62 
8 
71 
75 
9 
86 
79 
10 
43 
41 
11 
51 
41 
12 
42 
50 
13 
51 
47 
14 
55 
85 
15 
52 
39 
16 
47 
56 
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).
b. Explain why a pairedsamples t test is appropriate for answering this research question.
c. Based on the literature, the researcher is predicting a decline of memory functions in these elderly residents so the hypothesis is directional. What would be the null and alternative hypotheses in both words and symbol notations?
d. Calculate the difference score by subtracting each “Age 65” score from the associated “Age 75” score. Fill in the column in the table below for “difference score.”
Subject 
Difference score (Age 75 – Age 65) 
1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 
e. Calculate the mean from the sample of difference scores
f. Estimate the standard deviation of the population of difference scores
g. Calculate the standard error (standard deviation of the sampling distribution)
h. Calculate the t statistic for the sample of difference scores
i. Specify whether the test is a onetailed or twotailed test based on the hypotheses from (b), 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.
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.
k. Interpret the result in 12 sentences to answer the research question (you may use the wording from the hypothesis or explain it in your own words)
l. Calculate the standardized effect size of this hypothesis test
Solution:
(a)
The dependent variable is "Memory Score"
The independent varible is "Subject"
(b)
A compansion of two different memory scores of different ages where the scores are applied
to the same subjects. That's why we have to use paired t test for this research question.
(c)
d = Mean of Age 75 — Mean of Age 65.
Null hypothesis:
That is, there is no decline in memory functions based on this sample.
Alternative hypothesis:
That is, there is decline in memory functions based on this sample.
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