Question

Challenge  A business is asking customers if they are satisfied with the service. The business uses three different sampling methods and the conclusions are shown. Which method do you agree​ with? Explain. Use pencil and paper. If the business makes a conclusion about its​ customers, how accurate is the​ judgment?
Method​ 1: Systematic ​sample; StartFraction 95 Over 200 EndFraction
said they are satisfied with the​ service, so about 50​% are satisfied.
Method​ 2: Simple random ​sample; StartFraction 403 Over 1 comma 000 EndFraction
said they are not satisfied with the​ service, so about 40​% are not satisfied.
Method​ 3: Convenience ​sample; StartFraction 302 Over 500 EndFraction
said they are satisfied with the​ service, so about 60​% are satisfied.
Question content area bottom
Part 1
Which method do you agree​ with? Explain.
A.
Method 1 since it is the best representation of the population and has the least bias.
B.
Method 2 since it is the best representation of the population and has the largest sample size.
C.
Method 3 since it is the best representation of the population and has the highest percent of satisfaction.
D.
All methods are a good representation of the population.

Answer 1

Answer: B.

Reason: Method 2 is a simple random sample of size 1,000, so it is the least biased and the most representative of the customer population and it has the largest sample size. Systematic sampling (Method 1, n=200) can be acceptable but is smaller and can suffer from periodicity bias; the convenience sample (Method 3, n=500) is prone to selection bias and is not reliable as a population estimate.

Accuracy of the judgment (Method 2): If about 403/1000 are not satisfied, that implies about 59.7% satisfied (≈60%). The standard error for p ≈ 0.60 is sqrt(p(1−p)/n) ≈ sqrt(0.24/1000) ≈ 0.0155 (1.55%), so a 95% margin of error ≈ 1.96·1.55% ≈ 3.0%. Thus the true satisfaction rate is roughly 60% ± 3% assuming the sample was truly random and there is no nonresponse or measurement bias.