Question

You are interested in constructing a 95% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 357 randomly selected caterpillars observed, 41 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 357 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:-

a. given that n = 357 , x = 41

proportion p = x/n = 41/357 = 0.1148

the value of 95% confidence from z table is 1.96

confidence interval formula for proportion

=> p +/- z * sqrt(p*(1-p)/n)

=> 0.1148 +/- 1.96 * sqrt(0.1148*(1-0.1148)/357)

=> 0.0817 and 0.1479

b. If many groups of 357 randomly selected caterpillars were observed, then a different confidence interval would be produced from each group. About 95 percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about 5 percent will not contain the true population proportion