QuestionJune 5, 2025

Stealing Automobiles A random sample of 32 months has a mean of 64 automobiles stolen per month. The population standard deviation was 7 automobiles obtained from a long -term study. Find the 96% confidence interval for the mean. Round intermediate calculations to two decimal places and your final answers to the nearest whole number.

Stealing Automobiles A random sample of 32 months has a mean of 64 automobiles stolen per month. The population standard deviation was 7 automobiles obtained from a long -term study. Find the 96% confidence interval for the mean. Round intermediate calculations to two decimal places and your final answers to the nearest whole number.
Stealing Automobiles A random sample of 32 months has a mean of 64 automobiles stolen per month. The population standard deviation was 7 automobiles obtained from a long -term study. Find the 96% confidence interval for the mean. Round intermediate calculations to two decimal places and your final answers to the nearest whole number.

Solution
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Answer

(61, 67) Explanation 1. Identify the critical value For a 96\% confidence interval, the critical value z_{\alpha/2} is found using a standard normal distribution table. Since 96\% confidence level corresponds to \alpha = 0.04, we have \alpha/2 = 0.02. The critical value z_{0.02} \approx 2.05. 2. Calculate the standard error The standard error (SE) is calculated using the formula **SE = \frac{\sigma}{\sqrt{n}}**, where \sigma = 7 and n = 32. Thus, SE = \frac{7}{\sqrt{32}} \approx 1.24. 3. Compute the margin of error The margin of error (ME) is given by **ME = z_{\alpha/2} \times SE**. Substituting the values, ME = 2.05 \times 1.24 \approx 2.54. 4. Determine the confidence interval The confidence interval is calculated as **\bar{x} \pm ME**, where \bar{x} = 64. Therefore, the interval is 64 \pm 2.54, which gives (61.46, 66.54). 5. Round the final answer Round the interval to the nearest whole number: (61, 67).

Explanation

1. Identify the critical value<br /> For a $96\%$ confidence interval, the critical value $z_{\alpha/2}$ is found using a standard normal distribution table. Since $96\%$ confidence level corresponds to $\alpha = 0.04$, we have $\alpha/2 = 0.02$. The critical value $z_{0.02} \approx 2.05$.<br /><br />2. Calculate the standard error<br /> The standard error (SE) is calculated using the formula **$SE = \frac{\sigma}{\sqrt{n}}$**, where $\sigma = 7$ and $n = 32$. Thus, $SE = \frac{7}{\sqrt{32}} \approx 1.24$.<br /><br />3. Compute the margin of error<br /> The margin of error (ME) is given by **$ME = z_{\alpha/2} \times SE$**. Substituting the values, $ME = 2.05 \times 1.24 \approx 2.54$.<br /><br />4. Determine the confidence interval<br /> The confidence interval is calculated as **$\bar{x} \pm ME$**, where $\bar{x} = 64$. Therefore, the interval is $64 \pm 2.54$, which gives $(61.46, 66.54)$.<br /><br />5. Round the final answer<br /> Round the interval to the nearest whole number: $(61, 67)$.
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