Construct a 95% confidence interval based on the information presented.

Steps to Solve

1. Calculate the Sample Proportion

The sample proportion ($\hat{p}$) is calculated by dividing the number of workers who indicated anger ($x = 125$) by the total number of workers surveyed ($n = 750$):

$$ \hat{p} = \frac{x}{n} = \frac{125}{750} = 0.1667 $$

2. Determine the Standard Error

The standard error ($SE$) of the sample proportion is calculated using the formula:

$$ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} $$

Substituting the values we found:

$$ SE = \sqrt{\frac{0.1667(1 - 0.1667)}{750}} = \sqrt{\frac{0.1667 \times 0.8333}{750}} \approx 0.0176 $$

3. Find the Z-Score for 95% Confidence Level

For a 95% confidence level, the Z-score is approximately 1.96.

4. Calculate the Margin of Error

The margin of error ($ME$) is calculated as follows:

$$ ME = Z \times SE $$

Substituting the values:

$$ ME = 1.96 \times 0.0176 \approx 0.0345 $$

5. Construct the Confidence Interval

The confidence interval is constructed using the formula:

$$ \hat{p} \pm ME $$

So we have:

$$ CI = 0.1667 \pm 0.0345 $$

Calculating the lower and upper bounds:

• Lower bound: $0.1667 - 0.0345 = 0.1322$
• Upper bound: $0.1667 + 0.0345 = 0.2012$

Thus, the confidence interval is:

$$ CI = (0.1322, 0.2012) $$