We are interested in customer satisfaction ratings for a particular video game. In a
group of 65 customers, we �nd a sample mean of 42.95 and the standard deviation is 2.64.
Find a 95% Con�dence interval for u.
To find the 95% confidence interval for the population mean (µ), we can use the following formula:
Confidence interval = sample mean ± margin of error
The margin of error depends on the sample size, the standard deviation, and the desired level of confidence. In this case, we are given the following information:
Sample mean (x̄) = 42.95
Standard deviation (σ) = 2.64
Sample size (n) = 65
Level of confidence = 95%
First, we need to calculate the standard error of the mean (SE), which measures the average distance between the sample mean and the population mean. The formula for the standard error is:
SE = σ / √n
SE = 2.64 / √65 ≈ 0.3277
Next, we calculate the margin of error (ME), which represents the maximum expected difference between the sample mean and the population mean. The margin of error is calculated by multiplying the critical value (z*) by the standard error:
ME = z * SE
Since we want a 95% confidence interval, the critical value (z*) can be found from the standard normal distribution table. For a 95% confidence level, the z* value is approximately 1.96.
ME = 1.96 * 0.3277 ≈ 0.6425
Now, we can calculate the confidence interval by subtracting and adding the margin of error to the sample mean:
Confidence interval = sample mean ± margin of error
Confidence interval = 42.95 ± 0.6425
Confidence interval ≈ (42.31, 43.59)
Therefore, the 95% confidence interval for the population mean (µ) is approximately (42.31, 43.59).