Final answer:
To construct a 95% confidence interval for the mean age of customers, we use the sample mean of 33.40 years, the standard error calculated from the given standard deviation, and the critical t-value for 24 degrees of freedom. The resulting confidence interval is approximately between 29.51 years and 37.29 years.
Step-by-step explanation:
To construct a 95% confidence interval for the mean age of customers from a sample, we can use the formula:
Confidence interval = Sample mean ± (Critical value * Standard error)
In this case, the sample mean is given as 33.40 years, the standard deviation is 9.42 years, and the sample size is 25. The standard error (SE) is the standard deviation divided by the square root of the sample size (n).
SE = 9.42 / √25
= 9.42 / 5
= 1.884 years
For a 95% confidence interval with a sample size less than 30, we use the t-distribution. Assuming the sample was taken from a normally distributed population, at 24 degrees of freedom (n - 1), the critical t-value (two-tailed) can be found in a t-distribution table or with statistical software.
Assuming the t-value is approximately 2.064 (this value can vary slightly depending on the source), we calculate the margin of error (ME).
ME = Critical value * SE
= 2.064 * 1.884 years
≈ 3.89 years
Therefore, the 95% confidence interval is:
33.40 years ± 3.89 years
This means we are 95% confident that the true mean age of all the customers lies between approximately 29.51 years and 37.29 years.