Chapter 12, Problem AP4.42CPT

(a)

To determine

To state an appropriate 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 at that time.

Answer to Problem AP4.42CPT

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_1:p_1\ne p_2\end{array}$

Explanation of Solution

Now, we have to state an appropriate 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 at that time. Thus, the null hypothesis states that the proportions are equal i.e.

  $H_0:p_1=p_2$

And the alternative hypothesis states that the claim that the first hypothesis is less than the second hypothesis i.e.,

  $H_1:p_1\ne p_2$

(b)

To determine

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

Answer to Problem AP4.42CPT

Two sample z-test

Explanation of Solution

The hypotheses in above part (a) is as:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_1:p_1\ne p_2\end{array}$

Thus, we want to find out what inference method would you use to test the hypotheses you identified in part (a). Therefore, the population standard deviations are unknown and we compare two population proportions thus, we have that the corresponding test is then the two-sample z-test. Thus, two sample z-test will be used. We have assumed that the conditions for performing the inferences are met.

(c)

To determine

To explain can we generalize the results of this study to all women aged $65$ and older.

Answer to Problem AP4.42CPT

No, it cannot be generalized.

Explanation of Solution

The hypotheses in above part (a) is as:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_1:p_1\ne p_2\end{array}$

A study was conducted to know that can physical activity in youth lead to mental sharpness in old age or not. Thus, the result cannot be generalize of this study to all women aged $65$ and older because the samples are mostly based on white women, which do not represent all women. Thus, it is biased towards the true population.

(d)

To determine

To explain the concept of confounding and give the example of a potential confounding variable in this study.

Answer to Problem AP4.42CPT

A possible confounding variable is: Mentally stimulating activities.

Explanation of Solution

The hypotheses in above part (a) is as:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_1:p_1\ne p_2\end{array}$

A study was conducted to know that can physical activity in youth lead to mental sharpness in old age or not. Thus, we have two variables that are confounded when their effects on a response variable cannot be distinguished from each other. Therefore, a possible confounding variable is: Mentally stimulating activities.