Question

You are interested in constructing a 99% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 436 randomly selected caterpillars observed, 47 lived to become butterflies.

a. With 99% confidence the proportion of all caterpillars that lived to become a butterfly is between and .

b. If many groups of 436 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,

a) Point estimate = sample proportion = = x / n = 47 / 436 = 0.1078

1 - = 1 - 0.1078 = 0.8922

Z/2 = Z0.005 = 2.576

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

= 2.576 (((0.1078 * 0.8922) / 436 )

= 0.0383

A 99% confidence interval for population proportion p is ,

± E   

= 0.1078  ± 0.0383

= ( 0.0695, 0.1461 )

With 99% confidence the proportion of all caterpillars that lived to become a butterfly is between 0.0695 and 0.1461.

b. If many groups of 436 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