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Question

You work for a marketing firm that has a large client in the automobile industry. You have been asked to estimate the proportion of households in Chicago that have two or more vehicles. You have been assigned to gather a random sample that could be used to estimate this proportion to within a 0.035 margin of error at a 80% level of confidence.

a) With no prior research, what sample size should you gather in order to obtain a 0.035 margin of error? Round your answer up to the nearest whole number.

n =  households

b) Your firm has decided that your plan is too expensive, and they wish to reduce the sample size required. You conduct a small preliminary sample, and you obtain a sample proportion of ˆp=0.225p^=0.225 . Using this new information. what sample size should you gather in order to obtain a 0.035 margin of error? Round your answer up to the nearest whole number.

n =  households

Answer 1

Good afternoon and thank you for the question!

The answers are:

a) n = 335 houses

b) n = 234 houses

Below is an explanation of how I arrived at answers to parts a) and b):

a) We have no prior research to aid our estimate, so we want to sample the largest number of houses possible to get a margin of error of 0.035. For proportions, the equation for the margin of error, MOE, is:

MOE = z*√phat*(1-phat)/n.

In the above equation, phat is our point estimate for the proportion of households in Chicago that have two or more vehicles, z comes from the standard normal distribution and it's value is based on how confident we are in our point estimate: in this case, 80%, and n is the desired sample size.

Rearranging the above equation to solve for n, we get:

n = phat*(1-phat)*z²/MOE².

Again, we have no prior research, so we want to assume a value of phat=0.5 to give the largest sample size given a confidence level of 80% and a MOE of 0.035.

Note: If you play around with different values of phat, you will see phat=0.5 gives the largest value for phat*(1-phat).

For an 80% confidence level, z = 1.28. Now, we can plug all known values into the equation for n to get:

n = 0.5*(0.5)*1.28²/0.035² ≈ 334.37. With sample sizes, we always round up to the nearest integer, so we would need n = 335 houses.

b) Now, we have some prior research to refine our estimate for n. In this case, we know that phat = 0.225. We still want a MOE of 0.035 and we want to be 80% confident in our point estimate, phat. Therefore, we use the same formula for n as part a) to get:

0.225(1-0.225)*1.28²/0.035² ≈ 233.22, so we need a sample size of n = 234 houses.