Question

You are interested in constructing a 95% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 413 randomly selected caterpillars observed, 47 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 413 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

Solution :

Given that,

n = 413

x = 47

Point estimate = sample proportion = = x / n = 47 / 413 = 0.1138

1 - = 1 - 0.1138 = 0.8862

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.960}

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.96 (((0.1138 * 0.8862) / 413)

= 0.0306

A 95% confidence interval for population proportion p is ,

± E

= 0.1138  ± 0.0306

= ( 0.0832, 0.1444 )

correct option is = a.

With 95% confidence the proportion of all caterpillars that lived to become a butterfly is between 0.0832 and 0.1444.