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Elementary Statistics
Found in: Page 195
Elementary Statistics

Elementary Statistics

Book edition 9th
Author(s) Weiss, Neil
Pages 590 pages
ISBN 9780321989505

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Short Answer

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 of the format \(\hat p{\rm{ }} \pm {\rm{ }}E\)is\(0.25 \pm 0.071\).

See the step by step solution

Step by Step Solution

Step 1: Given information

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

Step 2: 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, the lower 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.

Step 3: Find the value of margin of error

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.

Step 4: 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:

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

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

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