(1)
You are interested in constructing a 99\% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 395 randomly selected caterpillars observed, 60 lived to become butterflies. Round answers to 4 decimal places where possible.
a. With $99 \%$ confidence the proportion of all caterpillars that lived to become a butterfly is between and
b. If many groups of 395 randomly selected caterpillars were observed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about percent will not contain the true population proportion.
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Answer
If many groups of 395 randomly selected caterpillars were observed, then a different confidence interval would be produced from each group. About 99 percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about 1 percent will not contain the true population proportion.
Step 1 ：Given that we have a sample size (n) of 395 caterpillars, and 60 of them lived to become butterflies. We are interested in constructing a 99% confidence interval for the proportion of all caterpillars that eventually become butterflies.
Step 2 ：First, we calculate the sample proportion (\(\hat{p}\)) which is the proportion of caterpillars that became butterflies in the sample. This is given by the number of successes (x) divided by the sample size (n). So, \(\hat{p} = \frac{x}{n} = \frac{60}{395} = 0.1519\).
Step 3 ：Next, we find the Z-score corresponding to the desired confidence level. For a 99% confidence level, the Z-score is approximately 2.576.
Step 4 ：We then calculate the standard error (SE) using the formula \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.1519(1-0.1519)}{395}} = 0.0181\).
Step 5 ：Finally, we calculate the confidence interval using the formula \(\hat{p} \pm Z \times SE\). The lower limit of the confidence interval is \(0.1519 - 2.576 \times 0.0181 = 0.1054\) and the upper limit is \(0.1519 + 2.576 \times 0.0181 = 0.1984\).
Step 6 ：\(\boxed{\text{Final Answer: With 99% confidence the proportion of all caterpillars that lived to become a butterfly is between 0.1054 and 0.1984.}}\)
Step 7 ：If many groups of 395 randomly selected caterpillars were observed, then a different confidence interval would be produced from each group. About 99 percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about 1 percent will not contain the true population proportion.
