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

The confidence interval for p is\(0.179 < p < 0.321\).

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 0.179 and upper confidence limit is 0.321.

Substituting values,

\(\begin{array}{c}\hat p = \frac{{0.3Step 3: Find the value of margin of error21 + 0.179}}{2}\\ = 0.25\end{array}\)

Hence, the sample proportion is 0.25.

Formula for margin of error is:

\(\begin{array}{c}E = \frac{{{\rm{upper confidence limit}} - {\rm{lower confidence limit}}}}{2}\\ = \frac{{0.321 - 0.179}}{2}\\ = 0.071\end{array}\)

Hence, the margin of error is 0.071.

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

\(\begin{array}{c}\hat p - E < p < \hat p + E\\0.25 - 0.071 < p < 0.25 + 0.071\\0.179 < p < 0.321\end{array}\)

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