Question

You are interested in constructing a 95% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 396 randomly selected caterpillars observed, 58 lived to become butterflies. Round answers to 4 decimal places where possible.a. With 95% confidence the proportion of all caterpillars that lived to become a butterfly is between and .b. If many groups of 396 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.

Answer 1

- The formula for proportion confidence interval is given as:

`\begin{gathered} p\pm Z^*\sqrt{(p(1-p))/(n)} \\ where, \\ Z^*\text{ is the z-score for the confidence level} \\ p\text{ is the proportion mean} \\ n\text{ is the sample size} \end{gathered}`

- Thus, we can find the confidence interval as follows:

Question A:

`\begin{gathered} p=(58)/(396)=0.1\overline{46} \\ \\ n=396 \\ Z^*=1.96 \\ \\ CI=0.1\overline{46}\pm1.96\sqrt{\frac{0.1\overline{46}(1-0.1\overline{46}}{396}} \\ \\ CI=0.1\overline{46}\pm0.03462456 \\ \\ CI=[0.111640086359,0.181289206571] \\ \\ CI=[0.1116,0.1813]\text{ \lparen To 4 decimal places\rparen} \end{gathered}`

Question B:

- If many groups of 396 randomly selected caterpillars are observed, about 95% will contain the true population proportion and about 5% will not contain the true population proportion