(Solved): It is known that the amount of time needed to change oil in a car is normally distributed with a st ...
It is known that the amount of time needed to change oil in a car is normally distributed with a standard deviation of 8 minutes. The amount of time to complete a random sample of 10 oil changes was recorded and listed as follows: 12111718191324221923 a) Compute the 99% confidence interval of the population mean. b) Compute the 90% confidence interval of the population mean. c) Describe the effect on the confidence interval when decreasing the confidence level. (3) (1)
a) To compute the 99% confidence interval of the population mean, we can use the t-distribution since the sample size is small (n = 10) and the population standard deviation is unknown. The formula for the confidence interval is: where X? is the sample mean, t is the critical value from the t-distribution for the desired confidence level and degrees of freedom (n-1), s is the sample standard deviation, and n is the sample size.First, let's calculate the sample mean and sample standard deviation:
Sample mean (X?) = Sample standard deviation (s) = Next, we need to find the critical value for the t-distribution with 9 degrees of freedom and a confidence level of 99%. Using a t-table or statistical software, we find the critical value to be approximately 3.250.Now we can calculate the confidence interval:
Confidence Interval =
Confidence Interval ? Therefore, the 99% confidence interval for the population mean is approximately .here we calculate the population mean.