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.