High School

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. The null and alternative hypotheses should be directional. The alpha level was set at [tex]\alpha = 0.05[/tex] 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 |

Tasks:

a. Identify the dependent variable (DV) and the independent variable (IV).
- **DV:** Memory score
- **IV:** Age group (65 vs. 75 years old)

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

c. 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 for each subject. Fill in the column for "Difference Score."

| 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.

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. Figure out the degrees 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 1-2 sentences to answer the research question.

l. Calculate the standardized effect size of this hypothesis test.

Answer :

Final answer:

The independent variable in this study is age and the dependent variable is the memory score. A paired-samples t-test is ideal for this research as we're measuring changes within the same subjects overtime. The assumptions of the null and alternative hypotheses, calculation of difference scores, mean, standard deviation, standard error, t-statistics, critical t value, degree of freedom, and standardized effect size have also been outlined.

Explanation:

In this research project we're looking at the observed differences in cognitive functions, specifically memory, of an elderly population over a period of 10 years. The dependent variable is the memory score while the independent variable is the age (65 vs 75 years old). A paired-samples t-test is appropriate here because we are assessing the changes in memory scores within the same subjects over time.

The alternative hypotheses in words would be that the mean difference in memory scores between age 65 and age 75 is less than 0 (indicating a decline); in symbolic notation: μd < 0. The null hypothesis would be μd = 0, suggesting no significant difference in memory function between these ages.

The difference scores (Age 75 – Age 65), mean of difference score, standard deviation, standard error and t-statistic would all need computations to identify. In terms of 'degree of freedom', it would be calculated as (N-1), where N represents the sample size. The critical t value(s) would be found in a t-distribution table, in this case, for a one-tailed test with the applicable degree of freedom and alpha level of 0.05.

If the calculated t-statistic is more extreme (either in the positive or negative direction) than the critical t-value, we would reject the null hypothesis; otherwise, fail to reject it. The final interpretation of the result should directly answer the researcher’s hypothesis. Lastly, the standardized effect size could be computed by dividing the t-statistic by the square root of the sample size.

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