Question

A store surveys customers to see if they are satisfied with the service they received. Samples of 25 surveys are taken. One in five people is unsatisfied. What is the variance of the mean of the sampling distribution of sample proportions for the number of unsatisfied customers? What is the variance for satisfied customers?

a) 0.16 and 0.04
b) 0.25 and 0.20
c) 0.20 and 0.25
d) 0.04 and 0.16

Answer 1

The variance for satisfied customers are b) 0.25 and 0.20.

Step-by-step explanation:

The variance of the sampling distribution of sample proportions for the number of unsatisfied customers (\(p\)) is calculated using the formula:

Var(p) = p . (1 - p)/n

where (p) is the proportion of unsatisfied customers, and (n) is the sample size. In this case, (p = 1/5) and (n = 25).

Var(p) = 1/5.(1 - 1/5)/25 = 4/125 = 0.032

Similarly, for satisfied customers (q = 1 - p), the variance can be calculated as:

Var(q) = q .(1 - q)/n

Var(q) = 4/5. (1 - 4/5)/25} = 16/125 = 0.128

Therefore, the variances for unsatisfied and satisfied customers are 0.032 and 0.128, respectively. To obtain the variance of the mean of the sampling distribution, divide these values by the sample size (n = 25):

`\[Var(\bar{p})` = 0.032/25 = 0.00128

`\[Var(\bar{q})` = 0.128/25 = 0.00512

So, the final variances for the mean of the sampling distribution of sample proportions for unsatisfied and satisfied customers are 0.00128 and 0.00512, respectively. In summary, the correct option is (b) 0.25 and 0.20.