Question

Formats of Confidence Intervals.

In Exercises 9–12, express the confidence interval using the indicated format. (The confidence intervals are based on the proportions of red, orange, yellow, and blue M&Ms in Data Set 27 “M&M Weights” in Appendix B.)

Orange M&Ms Express 0.179 < p < 0.321 in the form of $\hat{p}\pm E$.

Short answer

The confidence interval of the format $\hat{p}\pm E$is$0.25\pm 0.071$.

Step-by-step solution

Given information

The confidence interval for p is$0.179<p<0.321$

Find the value of sample proportion

Formula for sample proportion is:

$\hat{p}=\frac{\text{upper confidence limit}+\text{lower confidence limit}}{2}$

From the given confidence interval, thelower confidence limit is 0.179 and upper confidence limit is 0.321.

Substituting values,

$$\begin{gathered} \hat{p}=\frac{0.321+0.179}{2} \\ =0.25 \end{gathered}$$

Hence, the sample proportion is 0.25.

Find the value of margin of error

Formula for margin of error is:

$$\begin{gathered} E=\frac{\text{upper confidence limit}-\text{lower confidence limit}}{2} \\ =\frac{0.321-0.179}{2} \\ =0.071 \end{gathered}$$

Hence, the margin of error is 0.071.

Construct the confidence interval in the form of p^ ± E

The confidence interval in the form of $\hat{p}\pm E$is given as:

$$\begin{gathered} \hat{p}-E<p<\hat{p}+E \\ 0.25-0.071<p<0.25+0.071 \\ 0.179<p<0.321 \end{gathered}$$

Therefore, the confidence interval in the format $\hat{p}\pm E$is $0.25\pm 0.071$.