Chapter 7.3, Problem 28P

(a)

To determine

Find the 95%confidence interval for p using the plus four method.

Answer to Problem 28P

The 95%confidence interval for p using the plus four methodis $0.2763<p<0.5385$.

Explanation of Solution

Calculation:

Confidence interval:

The confidence interval formula is,

$\tilde{p}-E<p<\tilde{p}+E$

In the formula $E=z_c\sqrt{\frac{\tilde{p}\tilde{q}}{n+4}}$, $\tilde{p}$ is probability of success, $\tilde{q}$ is probability of failure, n is the sample size, c is confidence level, and $z_c$ is critical value for confidence level c based on the standard normal distribution.

Arandom sample of 50 trials with 20 successes is considered. That is, $n=50$, $r=20$.

The plus four point estimates for p is,

$\begin{array}{c}\tilde{p}=\frac{r+2}{n+4}\\ =\frac{20+2}{50+4}\\ =\frac{22}{54}\\ =0.4074\end{array}$

The confidence level is 95%

Critical value:

Use the Appendix II: Tables, Table 5 (b): Confidence interval, Critical Values $z_c$.

• In the level of confidence c column, locate the value of 0.95, or 95%.
• The corresponding Critical value $z_c$ is 1.96.

The error is,

$\begin{array}{c}E=1.96\sqrt{\frac{\left(0.4074\right)\left(1-0.4074\right)}{50+4}}\\ =1.96\times 0.0669\\ =0.1311\end{array}$

Confidence interval:

Substitute $\tilde{p}$ as 0.4074, E as 0.1311 in the confidence interval formula

$\begin{array}{c}0.4074-0.1311<p<0.4074+0.1311\\ 0.2763<p<0.5385\end{array}$

Hence, the 95%confidence interval for p using the plus four methodis $0.2763<p<0.5385$.

(b)

To determine

Find the 95% confidence interval for p.

Answer to Problem 28P

The 95% confidence interval for p is $0.2642<p<0.5358$.

Explanation of Solution

Calculation:

Confidence interval:

The confidence interval formula is,

$\widehat{p}-E<p<\widehat{p}+E$

In the formula $E=z_c\sqrt{\frac{\widehat{p}\widehat{q}}{n}}$, $\widehat{p}$ is probability of success, $\widehat{q}$ is probability of failure, n is the sample size, c is confidence level, and $z_c$ is critical value for confidence level c based on the standard normal distribution.

The point estimate for p is,

$\begin{array}{c}\widehat{p}=\frac{r}{n}\\ =\frac{20}{50}\\ =0.40\end{array}$

The error is,

$\begin{array}{c}E=1.96\sqrt{\frac{\left(0.4\right)\left(1-0.4\right)}{50}}\\ =1.96\times 0.0693\\ =0.1358\end{array}$

Confidence interval:

Substitute $\widehat{p}$ as 0.4, E as 0.1358 in the confidence interval formula

$\begin{array}{c}0.4-0.1358<p<0.4+0.1358\\ 0.2642<p<0.5358\end{array}$

Hence, the 90% confidence interval for the population proportion ofsuccesses p is $0.2642<p<0.5358$.

(c)

To determine

Compare the lengths of the confidence intervals of two methods.

Identify whether the point estimate is closer to $\frac{1}{2}$ when using the plus four method.

Identify whether the margin of error is smaller when using the plus four method.

Explanation of Solution

From part (a), 95% confidence interval using plus four methodis $\left(0.2763,0.5385\right)$. The length of interval is,

$\begin{array}{c}\text{length}=\text{Upper}\text{limit}-\text{lower}\text{limit}\\ =0.5385-0.2763\\ =0.2622\end{array}$

From part (b), 95% confidence intervalusing traditional method is $\left(0.2642,0.5358\right)$. The length of interval is,

$\begin{array}{c}\text{length}=\text{Upper}\text{limit}-\text{lower}\text{limit}\\ =0.5358-0.2642\\ =0.2716\end{array}$

It can be observed that the length of confidence interval using plus four method is shorter when compared to the length of confidence interval using traditional method.

From part (a), the point estimate using plus four method is 0.4074 and from part (b), the point estimate using traditional method is 0.40. It can be observed that, the point estimate using plus four method is closer to 0.50 when compared to traditional method.

Hence, the point estimate is closer to $\frac{1}{2}$ when using the plus four method.

From part (a), the margin of error using plus four method is 0.1311 and from part (b), the margin of error using traditional method is 0.1358. It can be observed that, the margin of error using plus four method is smaller when compared to traditional method

Hence, the margin of error is smaller when using the plus four method.