Question

Can physical activity in youth lead to mental sharpness in old age? A $2010$study investigating this question involved$9344$randomly selected, mostly white women over age $65$from four U.S. states. These women were asked about their levels of physical activity during their teenage years, $30$s,$50$ s, and later years. Those who reported being physically active as teens enjoyed the lowest level of cognitive decline-only $8.5\%$ had cognitive impairment-compared with $16.7\%$ of women who reported not being physically active at that time.
(a) State an appropriate pair of hypotheses that the researchers could use to test whether the proportion of women who suffered a cognitive decline was significantly smaller for women who were physically active in their youth than for women who were not physically active at that time. Be sure to define any parameters you use.
(b) Assuming the conditions for performing inference are met, what inference method would you use to test the hypotheses you identified in part (a)? Do not carry out the test.
(c) Suppose the test in part (b) shows that the proportion of women who suffered a cognitive decline was significantly smaller for women who were physically active in their youth than for women who were not physically active at that time. Can we generalize the results of this study to all women aged$65$ and older? Justify your answer.
(d) We cannot conclude that being physically active as a teen causes a lower level of cognitive decline for women over $65$, due to possible confounding with other variables. Explain the concept of confounding and give an example of a potential confounding variable in this study.

Short answer

(a)$\begin{array}{l}{H}_0:p_1=p_2\\ H;p_1\ne P_2\end{array}$

(b) Z - Test 2 sample.

(c) No, it's not possible to generalize it.

(d) Mentally stimulating activities could be a confusing variable.

Step-by-step solution

Part (a) Step 1: Given information

To develop a set of hypotheses that researchers may utilise to see if the proportion of women who experienced cognitive decline was lower among those who were physically active at the time.

Part (a) Step 2: Explanation

Now we need to develop a set of hypotheses that researchers can utilise to see if the proportion of women who had cognitive impairment was much lower among those who were physically active at the time. As a result, the null hypothesis argues that the proportions are the same

$H_0:p_1=p_2$

The alternative hypothesis asserts that the first hypothesis is less than the second hypothesis, i.e.

$H_1:p_1\ne p_2$

Part (b) Step 1: Given information

To explain what method of reasoning you would use to test the hypotheses you identified in part (a).

Part (b) Step 2: Explanation

The hypotheses in part (a) are as follows:

$$\begin{gathered} H_0:p_1=p_2 \\ {H}_1:p_1\ne p_2 \end{gathered}$$

As a result, we'd like to know which inference method you'd use to evaluate the hypotheses you selected in part 1. (a). As a result, we don't know what the population standard deviations are, therefore we compare two population proportions. As a result, we have the two-sample z-test as the appropriate test. As a result, two z-test samples will be used. The prerequisites for performing the inferences have been considered to be met.

Part (c) Step 1: Given information

To explain, can we apply the findings of this study to all women aged 65 and up

Part (c) Step 2: Explanation 

The hypotheses in part (a) are as follows:

$$\begin{gathered} H_0:p_1=p_2 \\ {H}_1:p_1\ne p_2 \end{gathered}$$
A study was carried out to see if physical activity in youth can lead to mental sharpness in later life. As a result, the findings of this study cannot be applied to all women aged 65 and up because the samples are largely made up of white women, who do not represent all women. Thus, it is biased towards the true population.

Part (d) Step 1: Given information

To define confounding and provide an example of a potential confounding variable in this study.

Part (d) Step 2: Explanation

The hypotheses in part (a) are as follows:

$$\begin{gathered} H_0:p_1=p_2 \\ {H}_1:p_1\ne p_2 \end{gathered}$$

A study was carried out to see if physical activity in youth can lead to mental sharpness in later life. When the effects of two factors on a response variable cannot be separated from one another, we have two variables that are confused. As a result, mentally stimulating activities could be a confusing variable.