Chapter 8, Problem 56SE

a.

To determine

Find the point estimate of the proportion of the population of voters rated as economy is good and excellent.

Answer to Problem 56SE

The point estimate of the proportion of the population of voters rated as economy is good and excellent is 0.22.

Explanation of Solution

Calculation:

The given information is that the sample of 750 voters participated in a survey about the election. The responses on state economy are collected. The 165 respondents rated as economy as good and excellent and 315 respondents rated as economy as poor.

The value of $\overline{p}$ is,

$\begin{array}{c}\overline{p}=\frac{\text{Numbe of possible cases}}{\text{Total number of cases}}\\ =\frac{165}{750}\\ =0.22\end{array}$

Thus, the point estimate of the proportion is 0.22.

b.

To determine

Find the 95% confidence interval for the proportion of voters rated as economy as good and excellent.

Answer to Problem 56SE

The 95% confidence interval for the proportion of voters rated as economy as good and excellent is (0.1904, 0.2496).

Explanation of Solution

Calculation:

The 95% confidence interval for proportion is,

$\overline{p}\pm z_{\frac{\alpha }{2}}\sqrt{\frac{\overline{p}\left(1-\overline{p}\right)}{n}}$ .

From the “Table 8.1 value of $z_{\frac{\alpha }{2}}$  ”, the value of z for 95% is 1.96.

The value of 95% confidence interval for population proportion is,

$\begin{array}{c}\overline{p}\pm z_{\frac{\alpha }{2}}\sqrt{\frac{\overline{p}\left(1-\overline{p}\right)}{n}}=0.22\pm \left(1.96\times \sqrt{\frac{0.22\left(1-0.22\right)}{750}}\right)\\ =0.22\pm \left(1.96\times \sqrt{\frac{0.1716}{750}}\right)\\ =0.22\pm \left(1.96\times 0.0151\right)\end{array}$

                             $\begin{array}{c}=0.22\pm 0.0296\\ =\left(0.22-0.0296,0.22+0.0296\right)\\ =\left(0.1904,0.2496\right)\end{array}$

Thus, the 95% confidence interval for population proportion is (0.1904, 0.2496).

c.

To determine

Find the 95% confidence interval for the proportion of voters rated as economy as poor.

Answer to Problem 56SE

The 95% confidence interval for the proportion of voters rated as economy as poor is (0.3847, 0.4553).

Explanation of Solution

Calculation:

The value of $\overline{p}$ is,

$\begin{array}{c}\overline{p}=\frac{\text{Numbe of possible cases}}{\text{Total number of cases}}\\ =\frac{315}{750}\\ =0.42\end{array}$

The 95% confidence interval for proportion is,

$\overline{p}\pm z_{\frac{\alpha }{2}}\sqrt{\frac{\overline{p}\left(1-\overline{p}\right)}{n}}$

From the “Table 8.1 value of $z_{\frac{\alpha }{2}}$  ”, the value of z for 95% is 1.96.

The value of 95% confidence interval for population proportion is,

$\begin{array}{c}\overline{p}\pm z_{\frac{\alpha }{2}}\sqrt{\frac{\overline{p}\left(1-\overline{p}\right)}{n}}=0.42\pm \left(1.96\times \sqrt{\frac{0.42\left(1-0.42\right)}{750}}\right)\\ =0.42\pm \left(1.96\times \sqrt{\frac{0.2436}{750}}\right)\\ =0.42\pm \left(1.96\times 0.018\right)\end{array}$

                             $\begin{array}{c}=0.42\pm 0.0353\\ =\left(0.42-0.0353,0.42+0.0353\right)\\ =\left(0.3847,0.4553\right)\end{array}$

Thus, the 95% confidence interval for population proportion is (0.3847, 0.4553).

d.

To determine

Find the wider confidence interval from part (b) and (c).

Answer to Problem 56SE

The confidence interval of part (c) is wider.

Explanation of Solution

Calculation:

From the results of part (b),

The 95% confidence interval for the proportion of voters rated as economy as good and excellent is (0.1904, 0.2496).

From the results of part (c),

The 95% confidence interval for the proportion of voters rated as economy as poor is (0.3847, 0.4553).

The confidence interval for part (c) is wider because the proportion 0.42 is closer to 0.5.

Thus, the confidence interval of part (c) is wider than part (b).