Final answer:
To construct a 90% confidence interval for the mean with a sample size of 92 and a known standard deviation of 9.5, we use the formula CI = X ± z * (s/√n). Rounding to two decimal places, the correct confidence interval is [35.22, 40.06]. Therefore, the correct option is a).
Step-by-step explanation:
To construct a confidence interval, we can use the formula:
CI = X ± z * (σ/√n)
Where CI is the confidence interval, X is the sample mean, σ is the population standard deviation, n is the sample size, and z is the z-score corresponding to the desired confidence level.
In this case, the sample mean X is unknown, so we use the formula:
CI = X ± z * (s/√n)
Where s is the sample standard deviation.
Given that the sample size n is 92, the standard deviation s is 9.5, and we want a 90% confidence interval, we can find the z-score by using a z-table or calculator. With a 90% confidence level, the z-score is approximately 1.645.
Now we can calculate the confidence interval:
CI = X ± 1.645 * (9.5/√92)
Rounding to two decimal places:
CI = X ± 1.645 * (0.992)
CI = X ± 1.63
Therefore, the correct option is a) A 90% confidence interval for the mean is [35.22, 40.06].