Chapter 22, Problem 23E

(a)

To determine

To Test: a suitable hypothesis and give the conclusion.

Explanation of Solution

Given:

  $\begin{array}{c}{n}_1=630\\ p_1=0.54\\ n_2=1010\\ p_2=0.51\end{array}$

Formula used:

  $S{E}_{Plooed}\left(p_1-p_2\right)=\sqrt{\frac{p_{Pooled}{q}_{Pooled}}{n_1}+\frac{p_{Pooled}{q}_{Pooled}}{n_2}}$

Calculation:

There is required to find out if there is clear evidence to show that due to the affair, there is a decline in voting support. By comparing two polls, this is done. Here's the premise

Let 1 denote the first poll and 2 denote all variables in the second poll.

  $H_0:p_1-p_2=0$

We want to test whether there was a reduction from the first poll to the second poll.

  $H_1:p_1-p_2>0$

Since the percentages are given, find $y_1$ and $y_2$

  $\begin{array}{c}{y}_1=n_1\times p_1\\ =630\times 0.54\\ =314\end{array}$

Now find $y_2$ :

  $\begin{array}{c}{y}_2=n_2\times p_2\\ =1010\times 0.51\\ =516\end{array}$

Now perform the following:

  $\begin{array}{c}{p}_{Plooed}=\frac{y_1+y_2}{n_1+n_2}\\ =\frac{314+516}{630+1010}\\ =0.523\end{array}$

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

  $\begin{array}{c}S{E}_{Plooed}\left(p_1-p_2\right)=\sqrt{\frac{p_{Pooled}{q}_{Pooled}}{n_1}+\frac{p_{Pooled}{q}_{Pooled}}{n_2}}\\ =\sqrt{\frac{0.523\times 0.477}{630}+\frac{0.523\times 0.477}{1010}}\\ =0.0254\end{array}$

Now, calculate the z value:

  $\begin{array}{c}z=\frac{\left(0.54-0.51\right)-0}{0.0254}\\ =1.18\end{array}$

Finally, using this value to find P in the table Z

  $P=P\left(z\ge 1.37\right)=0.119$

P-value is very big; therefore, it is failed to reject $H_0$ . There is no proof to denote there is a decrease in voter help due to the affair.

(b)

To determine

To explain: if the inference turns out to be incorrect, create a Type I or Type II error.

Answer to Problem 23E

Type II error

Explanation of Solution

Since the null hypothesis is incorrect and has not been discarded, it is a Type II error.

(c)

To determine

To calculate: the difference with an interval of confidence and describe the interval, if a difference was concluded.

Explanation of Solution

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