(a) A random survey of 100 mobile telephone customers indicated that 96 of them had not experienced any difficulties in making mobile calls in the previous 3 months.
(i) Derive a point estimate for customer satisfaction levels with the service
(ii) Construct a 95% Confidence Interval for the customer satisfaction level
(iii) In its submission to the Com-Reg regulator the phone company claims that 99% of its customers are satisfied with their service level. Use the results from (ii) above to comment on the company’s claim.
(i) To derive a point estimate for customer satisfaction levels with the service, we can take the proportion of customers who did not experience any difficulties in making mobile calls. In this case, the proportion is 96 out of 100 customers.
Point estimate for customer satisfaction level = Number of customers without difficulties / Total number of customers
= 96 / 100
= 0.96 or 96%
Therefore, the point estimate for customer satisfaction levels with the service is 96%.
(ii) To construct a 95% Confidence Interval for the customer satisfaction level, we can use the point estimate and the sample size along with the formula for calculating the confidence interval.
The formula for calculating the confidence interval is:
Confidence Interval = Point Estimate ± (Critical Value * Standard Error)
The critical value depends on the level of confidence desired and the sample size. For a 95% confidence level, the critical value can be found using a normal distribution table or a statistical calculator/software. In this case, the critical value is approximately 1.96.
The standard error can be calculated using the formula:
Standard Error = √((Point Estimate * (1 - Point Estimate)) / Sample Size)
Plugging in the values:
Point Estimate = 0.96
Sample Size = 100
Standard Error = √((0.96 * (1 - 0.96)) / 100)
= √(0.96 * 0.04 / 100)
= √0.0384 / 100
= √0.000384
≈ 0.0196 or 1.96%
Now, we can calculate the Confidence Interval:
Confidence Interval = 0.96 ± (1.96 * 0.0196)
= 0.96 ± 0.0384
= (0.9216, 0.9984)
Therefore, the 95% Confidence Interval for the customer satisfaction level is approximately (92.16%, 99.84%).
(iii) The company claims that 99% of its customers are satisfied with their service level. Comparing this claim to the confidence interval we calculated in (ii), we can see that the lower bound of the confidence interval is 92.16%.
Since the lower bound of the confidence interval is less than the company's claim of 99%, it suggests that the confidence interval does not support the company's claim. The data from the random survey indicates that the true customer satisfaction level may be lower than 99%. However, it is important to note that the confidence interval is an estimate with a certain level of uncertainty, so it is possible that the true satisfaction level could be higher or lower than the confidence interval.