An environmentalist wants to find out the fraction of oil tankers that have spills each month.
Step 2 of 2 : Suppose a sample of 281 tankers is drawn. Of these ships, 228 did not have spills. Using the data, construct the $90 \%$ confidence interval for the population proportion of oil tankers that have spills each month. Round your answers to three decimal places.
Answer
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Lower endpoint:
Upper endpoint:

Step 1 ：Given that a sample of 281 tankers is drawn and 228 of these ships did not have spills, we are to construct the $90 \%$ confidence interval for the population proportion of oil tankers that have spills each month.

Step 2 ：First, we calculate the sample proportion $\hat{p}$, which is the number of successes (tankers that did not have spills) divided by the sample size. In this case, $\hat{p} = \frac{228}{281} = 0.811$.

Step 3 ：The Z-score for a $90\%$ confidence level is approximately 1.645.

Step 4 ：The formula for a confidence interval for a proportion is given by $\hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $Z$ is the Z-score corresponding to the desired confidence level, and $n$ is the sample size.

Step 5 ：Substituting the values into the formula, we get the lower and upper endpoints of the confidence interval as $0.773$ and $0.850$ respectively.

Step 6 ：\(\boxed{\text{Final Answer: The } 90 \% \text{ confidence interval for the population proportion of oil tankers that have spills each month is approximately } [0.773, 0.850].}\)