The number of cars sold annually by used car salespeople is normally distributed with a standard deviation
of 19.22. A random sample of salespeople was taken and the number of cars each sold is listed her.
79 43 58 66 101 63 79 33
58 71 60 101 74 55 88
The 95% confidence interval estimate of the population mean is
(1) (58.873; 78.327)
(2) (66.099; 71.111)
(3) (686.000; 255.542)
(4) (68.600; 78.327)
(5) (588.730; 783.270)
To find the 95% confidence interval estimate for the population mean, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √n)
Where:
- Sample Mean is the average number of cars sold by the sample of salespeople
- Critical Value is obtained from the t-distribution table for a 95% confidence level and (n-1) degrees of freedom
- Standard Deviation is the standard deviation of the population (provided as 19.22)
- n is the sample size
Let's calculate the confidence interval:
Step 1: Calculate the Sample Mean
Add up all the car sales and divide by the number of salespeople in the sample:
79 + 43 + 58 + 66 + 101 + 63 + 79 + 33 + 58 + 71 + 60 + 101 + 74 + 55 + 88 = 1108
Sample Mean = 1108 / 15 = 73.87
Step 2: Calculate the Critical Value
The sample size is 15. Since the degrees of freedom is (n-1), we have 14 degrees of freedom. Looking up the t-distribution table for a 95% confidence level and 14 degrees of freedom, we find that the critical value is approximately 2.145.
Step 3: Calculate the Confidence Interval
Confidence Interval = 73.87 ± (2.145) * (19.22 / √15)
Confidence Interval = 73.87 ± (2.145) * (19.22 / 3.873)
Confidence Interval = 73.87 ± (2.145) * (4.979)
Confidence Interval = 73.87 ± 10.693
Confidence Interval ≈ (63.177, 84.563)
Therefore, the 95% confidence interval estimate for the population mean is approximately (63.177, 84.563).
None of the given answer choices match the calculated interval, so the correct choice is not provided.