Chapter 20, Problem 20.41E

(a)

To determine

To explain is there significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their forties.

Answer to Problem 20.41E

Yes, there is significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their forties.

Explanation of Solution

In the question, it is given that melatonin is a naturally occurring hormone involved in the regulation of body’s internal clock and is available as a dietary supplement in the United States. There is a table given for the treatment of melatonin and a placebo for women in forties and women in their fifties and sixties. Now, we want to know is there significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their forties. Thus, we define the hypotheses as:

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

That is, there is no statistical difference versus there is statistical difference.

The proportions are as:

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{10}{38}\\ =0.2632\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{20}{34}\\ =0.5882\end{array}$

Thus, to do the mechanics of a hypothesis test for equality of population proportions we will use the calculator $\text{TI}-89$ . Then select $6:2-$ PropZTest from the STAT Tests menu. Enter the observed counts and sample sizes. Indicate what kind of test you want: one-tail upper tail, lower tail, or two-tail. And specify whether results should simply be calculated or displayed with the area corresponding to the P-value of the test shaded.

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}z=2.79\\ p=0.0052\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we have sufficient evidence to conclude that there is significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their forties.

(b)

To determine

To explain is there significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their fifties and sixties.

Answer to Problem 20.41E

Yes, there is significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their fifties and sixties.

Explanation of Solution

In the question, it is given that melatonin is a naturally occurring hormone involved in the regulation of body’s internal clock and is available as a dietary supplement in the United States. There is a table given for the treatment of melatonin and a placebo for women in forties and women in their fifties and sixties. Now, we want to know is there significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their fifties and sixties. Thus, we define the hypotheses as:

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

That is, there is no statistical difference versus there is statistical difference.

The proportions are as:

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{22}{30}\\ =0.7333\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{18}{38}\\ =0.4737\end{array}$

Thus, to do the mechanics of a hypothesis test for equality of population proportions we will use the calculator $\text{TI}-89$ . Then select $6:2-$ PropZTest from the STAT Tests menu. Enter the observed counts and sample sizes. Indicate what kind of test you want: one-tail upper tail, lower tail, or two-tail. And specify whether results should simply be calculated or displayed with the area corresponding to the P-value of the test shaded.

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}z=2.16\\ p=0.0308\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we have sufficient evidence to conclude that there is significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their fifties and sixties.

(c)

To determine

To write a short description of your findings contrasting your results for part (a) and (b).

Explanation of Solution

In the question, it is given that melatonin is a naturally occurring hormone involved in the regulation of body’s internal clock and is available as a dietary supplement in the United States. There is a table given for the treatment of melatonin and a placebo for women in forties and women in their fifties and sixties. From, part (a) we have the result as there is significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their forties. And from part (b) we have the result as, there is significant evidence of a difference in the proportions of women with increased LH levels between women taking melatonin and women taking placebo, for women in their fifties and sixties. Thus, for the women is forties, the proportion with increased LH levels is significantly greater in the placebo group and for the women is fifties and sixties, the proportion with increased LH levels is significantly greater in the melatonin group. Therefore, the effect is reversed in the two age groups.

(d)

To determine

To show that these results are not statistically significant.

Answer to Problem 20.41E

We can conclude that there is no significant difference.

Explanation of Solution

In the question, it is given that melatonin is a naturally occurring hormone involved in the regulation of body’s internal clock and is available as a dietary supplement in the United States. There is a table given for the treatment of melatonin and a placebo for women in forties and women in their fifties and sixties. Thus, we define the hypotheses as:

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

That is, there is no statistical difference versus there is statistical difference.

The proportions are as:

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{32}{68}\\ =0.4706\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{38}{72}\\ =0.5278\end{array}$

Thus, to do the mechanics of a hypothesis test for equality of population proportions we will use the calculator $\text{TI}-89$ . Then select $6:2-$ PropZTest from the STAT Tests menu. Enter the observed counts and sample sizes. Indicate what kind of test you want: one-tail upper tail, lower tail, or two-tail. And specify whether results should simply be calculated or displayed with the area corresponding to the P-value of the test shaded.

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}z=0.68\\ p=0.4988\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P>0.05\Rightarrow \text{Fial to Reject }{H}_0$

Thus, we can conclude that there is no significant difference since the pooling of the two groups cancels out the two separate effects.