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\({\rm{\hat p \pm E}}\).

Short answer

The confidence interval in the form of\(\hat p{\rm{ }} \pm {\rm{ }}E\)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:

\(\hat p = \frac{{{\rm{upper confidence limit}} + {\rm{lower confidence limit}}}}{2}\)

From the given confidence interval, thelower confidence limit is 0.0434 and the upper confidence limit is 0.217.

Substituting values,

\(\begin{aligned}{c}\hat p = \frac{{0.217 + 0.0434}}{2}\\ = 0.1302\end{aligned}\)

Hence, the sample proportion is 0.130.

Find the value of margin of error

Formula for margin of error is:

\(\begin{aligned}{c}E = \frac{{{\rm{upper confidence limit}} - {\rm{lower confidence limit}}}}{2}\\ = \frac{{0.217 - 0.0434}}{2}\\ = 0.0868\end{aligned}\)

Hence, the margin of error is 0.087.

Construct the confidence interval in the form of \({\rm{\hat p  \pm   E}}\)

The confidence interval in the form of\(\hat p \pm E\)is given as:

\(\begin{aligned}{c}\hat p - E < p < \hat p + E\\0.1302 - 0.0868 < p < 0.1302 + 0.0868\\0.0434 < p < 0.217\end{aligned}\)

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