Question

Starbucks wants to know about the average age of their customers. A random sample of 22 customers is taken. It shows a mean age of 27 years with a standard deviation of 3 years. The width of a 90 percent CI for the true mean client age is approximately:

a) 1.47 years
b) 1.96 years
c) 2.33 years
d) 2.58 years

Answer 1

The width of a 90 percent confidence interval for the true mean client age is calculated using the formula + or - z * (standard deviation / sqrt(n)), where z is the z-value from the z-distribution table. However, the given options do not match the calculated width using the provided sample statistics, suggesting a possible typo in the options.

Step-by-step explanation:

The width of a 90 percent confidence interval for the true mean client age can be calculated using the formula for a confidence interval: ± z * (standard deviation / √n), where z is the z-value corresponding to the desired confidence level, the standard deviation (s) is the standard deviation of the sample, and n is the sample size.

For a 90% confidence interval and a sample size of 22, the appropriate z-value from the z-distribution table is about 1.645 (since it's a two-tailed test, we look for 0.95 in the cumulative probability, which corresponds to 0.05 in the tail). Therefore, with s = 3 and n = 22, the calculation will be:

1.645 * (3 / √22) ≈ 1.645 * (3 / 4.69) ≈ 1.645 * 0.639 ≈ 1.05 years

However, this does not match any of the options provided. It's possible there may be a typo in the provided options, as none of the choices are correct based on the given statistics. We would need to double-check the values and the work for errors.