Chapter 22, Problem 24E

(a)

To determine

To Test: a suitable hypothesis and give conclusion.

Explanation of Solution

Given:

  $\begin{array}{c}{n}_1=2222\\ p_1=0.21\\ n_2=208\\ p_2=0.18\end{array}$

Formula used:

  $\begin{array}{c}{p}_{Plooed}=\frac{y_1+y_2}{n_1+n_2}\\ S{E}_{Plooed}\left(p_1-p_2\right)=\sqrt{\frac{p_{Pooled}{q}_{Pooled}}{n_1}+\frac{p_{Pooled}{q}_{Pooled}}{n_2}}\end{array}$

Calculation:

There is required to find out why there is good evidence to show why men are more likely than women to buy books online. Here's the premise

Let 1 denote the man and 2 for the survey denote the women.

  $H_0:p_1-p_2=0$

There is required to test whether women are likely to buy less online books.

  $H_1:p_1-p_2>0$

  $\begin{array}{c}{y}_1=n_1\times p_1\\ =222\times 0.21\\ =47\end{array}$

Now find $y_2$ :

  $\begin{array}{c}{y}_2=n_2\times p_2\\ =208\times 0.18\\ =38\end{array}$

Now perform the following:

  $\begin{array}{c}{p}_{Plooed}=\frac{y_1+y_2}{n_1+n_2}\\ =\frac{47+38}{222+208}\\ =0.198\end{array}$

Next find the $S{E}_{Plooed}$

  $\begin{array}{c}S{E}_{Plooed}\left(p_1-p_2\right)=\sqrt{\frac{p_{Pooled}{q}_{Pooled}}{n_1}+\frac{p_{Pooled}{q}_{Pooled}}{n_2}}\\ =\sqrt{\frac{0.198\times 0.802}{222}+\frac{0.198\times 0.802}{208}}\\ =0.0384\end{array}$

Calculate the value of z is

  $\begin{array}{c}z=\frac{\left(0.21-0.18\right)-0}{0.0254}\\ =0.780\end{array}$

Using this value to find value of P in the Z table.

  $P=P\left(z\ge 0.78\right)=0.22$

This P-value, then, is very large; thus, there is fail to reject $H_0$ . There is no proof to suggest that men are more likely than women to buy books online.

(b)

To determine

To explain: if the conclusion is indeed false, create a Type I or Type II error.

Answer to Problem 24E

Type II error

Explanation of Solution

It is a Type II error since the null hypothesis is incorrect and it is failed to reject it.

(c)

To determine

To find: this difference with a confidence interval.

Explanation of Solution

There is no need to do an interval because it is failed to reject it. The difference will be calculated by this interval.