Question

a. For the cases below, calculate the standard error for estimate of the proportion?                                              

n = 500 and p = 0.1                                             

n = 100 and p = 0.92                       

b. Comment on whether the sample sizes are large enough so that the sample proportions can be approximated by a normal distribution

Answer 1

Part a)

The formula for the standard error for estimate of the proportion = sqrt(p*(1-p)/n)

For n = 500 and p = 0.1

The standard error for estimate of the proportion = sqrt(0.1*(1-0.1)/500)

The standard error for estimate of the proportion = 0.0134

For n = 100 and p = 0.92

The standard error for estimate of the proportion = sqrt(0.92*(1-0.92)/100)

The standard error for estimate of the proportion = 0.0271

Part b)

To check whether sample sizes are large enough following conditions should be met

n*p ≥ 10

n*(1-p) ≥ 10

For n = 500 and p = 0.1

500*0.1 = 50 ≥ 10 -> Condition Satisfied

500*(1-0.1) = 450 ≥ 10 -> Condition Satisfied

Thus, sample size is large enough

For n = 100 and p = 0.92

100*0.92 = 92 ≥ 10 -> Condition Satisfied

100*(1-0.92) = 8 < 10 -> Condition Not Satisfied

Thus, sample size is not large enough in this case.