Chapter 12: Q. 42 (page 807)

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 lower 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 (b)? 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 lower 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

Expert verified

(a) The given statement is proved below.

(b) The inference method would you use to test the hypotheses you identified in part (b) is the-sample-z-test.

(d) A possible confounding variable is: Mentally stimulating activities.

Step by step solution

Part (a) step 1: Given Information

We need to find an appropriate pair of hypotheses that the researchers could use to test whether the proportion of women who suffered a cognitive decline was significantly lower for women who were physically active in their youth than for women who were not physically active at that time.

Part (a) step 2: Explanation

The null hypothesis states the proportions are equal by:

$H_{0} : p_{1} = p_{2}$

The alternative hypothesis states the claim that the first proportion is less than the second proportion as:

$H_{a} : p_{1} < p_{2}$

Part (b) step 1: Given Information

We need to find an inference method would you use to test the hypotheses you identified in part (b).

Part (b) step 2: Explanation

The population standard deviations are unknown and we compare two population proportions, the corresponding test is then the-sample-z-test.

Part (c) step 1: Given Information

We need to find that the proportion of women who suffered a cognitive decline was significantly lower for women who were physically active in their youth than for women who were not physically active at that time.