Final answer:
Using the sample mean of 57.2 seconds, a population standard deviation of 11.3 seconds, and a sample size of 40 students, the 99% confidence interval for the population mean μ is calculated to be (55.48, 59.92). This interval is the result of applying the z-score for a 99% confidence level and accounting for the standard error of the mean.
Step-by-step explanation:
To calculate the 99% confidence interval for the population mean μ, we use the sample mean, the population standard deviation, and the appropriate z-value for the 99% confidence level. Given that the sample mean is 57.2 seconds, the population standard deviation (σ) is 11.3 seconds, and the sample size (n) is 40, we proceed to find the z-score for a 99% confidence level.
First, we find the z-score for 99% confidence level which corresponds to a 0.5% probability in each tail (since the normal distribution is symmetric). The z-score approximately equals 2.576.
Next, we calculate the standard error of the mean (SEM) using the formula: SEM = σ / √n. Substituting the given values, SEM = 11.3 / √40.
The margin of error (E) is then the z-score multiplied by the SEM: E = z * SEM. Plugging in the numbers, we calculate E.
Finally, the 99% confidence interval is found by subtracting and adding the margin of error from and to the sample mean: CI = sample mean ± E. After the calculations, we find that the confidence interval falls within the range of (55.48, 59.92).
Interpretation: We are 99% confident that the interval (55.48, 59.92) contains the true population mean μ of the students' ability to estimate when sixty seconds have passed.