Chapter 8.3, Problem 28P

Expand Your Knowledge: Plus Four Confidence Interval for a Single Proportion One of the technical difficulties that arises in the computation of confidence intervals for a single proportion is that the exact formula for the maximal margin of error requires knowledge of the population proportion of success p. Since p is usually not known, we use the sample estimate p = r/n in place of p As discussed in the article "How Much Confidence Should You Have in Binomial Confidence Intervals?"appearing in issue no, 45 of the magazine STATS (a publicationof the American Statistical Association), use of p as an estimate for p means that the actual confidence level for the intervals may in fact he smaller than the specified level c. This problem arises even when n is large, especially if p is not near 1/2.

A simple adjustment to the formula for the confidence intervals is the plus estimate, first suggested by Edwin Bidwell Wilson in 1927. It is also called the Agresti-Coull confidence interval. This adjustment works best for 95% confidence intervals

The plus four adjustment has us add two successes and two failures to the sample data. This means that r, the number of successes, is increased by 2, and n, the sample size, is increased by 4. We use the symbol $\tilde{p}$, read tilde," for the resulting sample estimate of P, So. $\tilde{p}=\left(r+ 2\right)/\left(n+4\right).$

(a) Consider a random sample of 50 trials with 20 successes. Compute a 95% confidence interval for p using the plus four method (b) Compute a traditional 95% confidence interval for p using a random sample of 50 trials with 20 successes.

(c) Compare the lengths of the intervals obtained using the two methods. Is the point estimate closer to 1/2 when using the plus four method? Is the margin of error smaller when using the plus four method?

To determine

To find: The95% confidence interval for p using the plus four method.

Answer to Problem 28P

Solution:

The 95% confidence interval for $p$ is (0.28, 0.54).

Explanation of Solution

Calculation:

Let r be a random variable that represents the number of successes out of the n binomial trials.

$r=20,n=50$

The plus four point estimates for p and q are,

$\begin{array}{l}\tilde{p}=\frac{r+2}{n+4}\\ \tilde{p}=\frac{20+2}{50+4}\\ \tilde{p}=\frac{22}{54}\\ \tilde{p}\approx 0.41\end{array}$

$\begin{array}{l}\tilde{q}=1-\tilde{p}\\ \tilde{q}=1-0.41\\ \tilde{q}\approx 0.59\end{array}$

We have to find 95% confidence interval,

$\begin{array}{l}c=0.95\\ \text{Using}\text{Table}\text{3}\text{of}\text{Appendix}\\ z_{0.95}=1.96\\ \tilde{p}=0.41\\ \tilde{q}=0.59\\ n=50\\ E=z_c\sqrt{\frac{\tilde{p}\tilde{q}}{n+4}}\\ E=1.96\sqrt{\frac{0.41(0.59)}{50+4}}\\ E=0.1312\\ E\approx 0.13\end{array}$

95% confidence interval is

$\begin{array}{l}\tilde{p}-E<p<\tilde{p}+E\\ 0.41-0.13<p<0.41+0.13\\ 0.28<p<0.54\end{array}$

The 95% confidence interval for $p$ is (0.28, 0.54).

Margin of error is $E=0.13$

To determine

To find: Thetraditional95% confidence interval for p using the plus four method.

Answer to Problem 28P

Solution:

The 95% confidence interval for $p$ is (0.28, 0.54).

Explanation of Solution

Calculation:

Let r be a random variable that represents the number of successes out of the n binomial trials.

$r=20,n=50$

We estimate p, q by the sample point estimates:

$\begin{array}{c}\widehat{p}=\frac{r}{n}\\ =\frac{20}{50}\\ \approx 0.4\end{array}$

$\begin{array}{c}\widehat{q}=1-\widehat{p}\\ =1-0.4\\ =0.6\end{array}$

We have to find 95% confidence interval,

$\begin{array}{l}c=0.95\\ n=50\\ \text{Using}\text{Table}\text{3}\text{of}\text{Appendix}\\ z_{0.95}=1.96\\ \widehat{p}=0.4\\ E=z_c\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\\ E=1.96\sqrt{\frac{0.4(1-0.4)}{50}}\\ E\approx 0.14\end{array}$

90% confidence interval is

$\begin{array}{l}\widehat{p}-E<p<\widehat{p}+E\\ 0.4-0.14<p<0.4+0.14\\ 0.26<p<0.54\end{array}$

The 95% confidence interval for $p$ is (0.26, 0.54).

Margin of error is $E=0.14$

To determine

To explain: Whether the point estimate is closer to $\frac{1}{2}$ and the margin of error is smaller when we used the plus four method.

Answer to Problem 28P

Solution: Yes, the point estimate is closer to $\frac{1}{2}$ and the margin of error is smaller when we used the plus four method.

Explanation of Solution

Since the95% confidence interval for p using the plus four method is (0.28, 0.54) and the95% confidence interval for p using a traditional method is (0.26, 0.54). The length of confidence interval using the plus four method is 0.26 and for traditional method it is 0.28. Hence, the length of the confidence interval using plus four method is slightly smaller than traditional method because the sample size is increased by 4 in plus four method.

The point estimate when using the plus four method is 0.41 which is nearly closer to $\frac{1}{2}$.

The margin of error is smaller when we used the plus four method because the sample size is increased by 4 in plus four method.