The proportion of supermarket customers who do not buy store-brand products is to be estimated. Which of the following scenarios would lead to a sampling distribution of the sample proportion with the lowest variability?

a Sample 100 customers from the roughly 2000 customers who shop at one store location.

b Sample 100 customers from the roughly 20,000 customers who shop at the stores citywide.

c Sample 200 customers from the roughly 2000 customers who shop at one store location.

d Sample 300 customers from the roughly 20,000 customers who shop at the stores citywide.

We have a finite population (one for the customers who shop at one store location and other bigger that is for the customers who shop at the stores citywide).

The standard error for a finite population can be written as:

[tex]\sigma_x=\frac{s}{\sqrt{n}} \sqrt{1-\frac{n}{N} }[/tex]

For each population, the higher the sample size, the less variablity will have in the estimation of the proportion. So, we are left with option C and D.

We can calculate the standard error for each posibility and compare:

c) Sample n=200 customers from the roughly N=2000 customers who shop at one store location.

[tex]\sigma_x=\frac{s}{\sqrt{100}}\sqrt{1-\frac{200}{2000} } \\\\ \sigma_x=\frac{s}{10} \sqrt{1-0.1 }=s(0.1*0.949)=0.0949s[/tex]

d) Sample 300 customers from the roughly 20,000 customers who shop at the stores citywide.

[tex]\sigma_x=\frac{s}{\sqrt{100}}\sqrt{1-\frac{300}{20000} } \\\\ \sigma_x=\frac{s}{10} \sqrt{1-0.015 }=s(0.1*0.992)=0.0992s[/tex]

It will give less variability the Option C.

It is assumed that sampling only one store is representative of the parameter of the population of study.