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Q10

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Found in: Page 195

### Elementary Statistics

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

# 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 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 $$0.25 \pm 0.071$$.