In a 2001 a Gallup poll surveyed 1016 households in the U S

In a 2001, a Gallup poll surveyed 1016 households in the U. S about their pets. Of those surveyed 599 said they had at least one dog or cat as a pet.

(a). Find and interpret a 90% CI for the population proportion.

(b). Find and interpret a 95% CI for the population proportion.

(c). Find and interpret a 99% CI for the population proportion.

(d). (i) Find the margin of errors from (a)-(c) and compare them.

(ii).Which interval has the smallest margin of error? and why?

(a). Find and interpret a 90% CI for the population proportion.

p=599/1016=0.5895669

Given a=1-0.9=0.1, Z(0.05) = 1.645 (from standard normal table)

So the lower bound is

p-Z*sqrt(p*(1-p)/n) =0.5895669 - 1.645*sqrt(0.5895669*(1-0.5895669)/1016) =0.5641802

So the upper bound is

p+Z*sqrt(p*(1-p)/n) =0.5895669 + 1.645*sqrt(0.5895669*(1-0.5895669)/1016) =0.6149536

We have 90% confident that the population proportion will be within this interval

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(b). Find and interpret a 95% CI for the population proportion.

Given a=0.05, Z(0.025) = 1.96 (from standard normal table)

So the lower bound is

p-Z*sqrt(p*(1-p)/n) =0.5895669 - 1.96*sqrt(0.5895669*(1-0.5895669)/1016) =0.5593189

So the upper bound is

p+Z*sqrt(p*(1-p)/n) =0.5895669 + 1.96*sqrt(0.5895669*(1-0.5895669)/1016) =0.6198149

We have 95% confident that the population proportion will be within this interval

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(c). Find and interpret a 99% CI for the population proportion.

Given a=0.01, Z(0.005) = 2.58 (from standard normal table)

So the lower bound is

p-Z*sqrt(p*(1-p)/n) =0.5895669 - 2.58*sqrt(0.5895669*(1-0.5895669)/1016) =0.5497506

So the upper bound is

p+Z*sqrt(p*(1-p)/n) =0.5895669 + 2.58*sqrt(0.5895669*(1-0.5895669)/1016) =0.6293832

We have 99% confident that the population proportion will be within this interval

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(d). (i) Find the margin of errors from (a)-(c) and compare them.

the margin of error for (a) is 1.645*sqrt(0.5895669*(1-0.5895669)/1016)=0.02538673

the margin of error for (b) is 1.96*sqrt(0.5895669*(1-0.5895669)/1016)=0.03024802

the margin of error for (c) is 2.58*sqrt(0.5895669*(1-0.5895669)/1016)=0.03981627

When the confidence level increases, the margin of error increases

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(ii).Which interval has the smallest margin of error? and why?

The 90% confidence interval has the smallest margin of error because the value of the confidence level is the smallest