Forty-nine items are randomly selected from a population of 500 items. The sample mean is 40 and the sample standard deviation 9. Develop a 99% confidence interval for the population mean. (Round your answers to 3 decimal places.)
To develop a 99% confidence interval for the population mean, we'll use the formula:
Confidence interval = sample mean ± (critical value * sample standard deviation / √sample size)
First, let's determine the critical value corresponding to a 99% confidence level. We can find this value by using a t-table or a statistical calculator. Since the sample size is 49, we have degrees of freedom (df) equal to 49 - 1 = 48.
Looking up the critical value using a t-table for a confidence level of 99% and 48 degrees of freedom, we find the critical value to be approximately 2.682.
Now, we can substitute the given values into the formula:
Confidence interval = 40 ± (2.682 * 9 / √49)
Calculating the expression inside the parentheses, we have:
Confidence interval = 40 ± (2.682 * 9 / 7)
Next, we simplify the expression:
Confidence interval = 40 ± (3.746 / 7)
Finally, we calculate the confidence interval:
Confidence interval = 40 ± 0.535
Rounding to three decimal places, we get:
Confidence interval = (39.465, 40.535)
Therefore, the 99% confidence interval for the population mean is (39.465, 40.535).