You are interested in constructing a 90% confidence interval for the proportion of all caterpillarsthat eventually become butterflies. Of the 402 randomly selected caterpillars observed, 42 livedto become butterflies. Round answers to 4 decimal places where possible.a. With 90% confidence the proportion of all caterpillars that lived to become a butterfly isbetweenandb. If many groups of 402 randomly selected caterpillars were observed, then a differentpercent of theseconfidence intervals will contain the true population proportion of caterpillars that becomeconfidence interval would be produced from each group. Aboutbutterflies and aboutpercent will not contain the true population proportion.

(a) Obtain the 90% confidence interval for the proportion of all caterpillars that eventually become butterflies. The 90% confidence interval for the proportion of all caterpillars that eventually become butterflies is obtained below: Use EXCEL Procedure for finding the critical value of z. Follow the instruction to obtain the critical value of z: Open EXCEL Go to Formula bar. In formula bar enter the function as“=NORMINV” Enter the probability as 0.95. Enter the mean as 0 and the Standard deviation as 1. Click enter EXCEL output: From the EXCEL output, the critical value of z at the 90% confidence level is 1.645.Thus, the 90% confidence interval for the proportion of all caterpillars that eventually become butterflies lies between 0.0794 and 0.1296. With 90% confidence the proportion of all caterpillars that lived to become a butterfly is between 0.0794 and 0.1296. (b) Determine the number of groups of 402 randomly selected caterpillars that were observed, then a different confidence interval would be produced from each group. It is clear that about 90%percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and only about 10% percent will not contain the true population proportion. Thus, About 90%percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about 10% percent will not contain the true population proportion.