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.)

Red M&Ms Express 0.0434 < p < 0.217 in the form of$p+E$

Short answer

The confidence interval in the form of $\hat{p}\pm E$is$0.130\pm 0.087$

Step-by-step solution

Given information

The confidence interval for p is$0.0434<p<0.217$

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 0.0434 and the upper confidence limit is 0.217.

Substituting values,

$$\begin{gathered} \hat{p}=\frac{0.217+0.0434}{2} \\ =0.1302 \end{gathered}$$

Hence, the sample proportion is 0.130.

Find the value of margin of error

Formula for margin of error is:

$$\begin{gathered} E=\frac{\text{upper confidence limit}-\text{lower confidence limit}}{2} \\ =\frac{0.217-0.0434}{2} \\ =0.0868 \end{gathered}$$

Hence, the margin of error is 0.087.

Construct the confidence interval in the form of p±E

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

$$\begin{gathered} \hat{p}-E<p<\hat{p}+E \\ 0.1302-0.0868<p<0.1302+0.0868 \\ 0.0434<p<0.217 \end{gathered}$$

Therefore, the confidence interval in the form of $\hat{p}\pm E$is $0.1302\pm 0.0868$.