Question

You are interested in constructing a 90% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 439 randomly selected caterpillars observed, 45 lived to 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 is between  and .

b. If many groups of 439 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

a)

sample success x =		45
sample size          n=		439
sample proportion p̂ =x/n=		0.1025
std error se= √(p*(1-p)/n) =		0.0145
for 90 % CI value of z=		1.645
margin of error E=z*std error   =		0.0238
lower bound=p̂ -E                       =		0.0787
Upper bound=p̂ +E                     =		0.1263

90% confidence the proportion of all caterpillars that lived to become a butterfly is between 0.0787 and 0.1263

b)

about 95% of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about 5% will not contain the true population proportion.