Question 1175516
sample size is 25
sample mean is 72 minutes
population standard deviation is 5 minutes.


standard error = standard deviation / square root of sample size = 5 / sqrt(25) = 5/5 = 1.


z-score = (x - m) / s


x is the raw score
m is the raw mean
s is the standard error.


at two tailed confidence level, the tails on each end of the confidence interval are 2.5%.


the critical z-score is therefore equal to plus or minus 1.96.


solve for the high raw score as follows:


start with 1.96 = (x - 72) / 1


solve for x to get x = 1.96 * 1 + 72 = 73.96.


solve for the low raw score as follows:


start with -1.96 = (x - 72) / 1


solve for x to get x = -1.96 * 1 + 72 = 70.04


your answer for question (a) is 70.04 to 73.96 minutes.


with a two tailed 97% confidence level, the tails on each end of the confidence interval are 1.5%.


the critical z-score becomes plus or minus 2.17.


solve for the low and high raw score in the same manner as you did the the 95% confidence interval.


you will get the low score equal to 69.83 and the high score equal to 74.17.


visually, this looks like this.


first 95% confidence interval, then 97% confidence interval.


<img src = "http://theo.x10hosting.com/2021/030701.jpg" >


<img src = "http://theo.x10hosting.com/2021/030702.jpg" >


when dealing with the mean of a sample, you do not use the standard deviation.


you use the standard error.


the standard error is defined as the standard deviation of the distribution of sample means.


here's a reference.


<a href = "https://www.investopedia.com/terms/s/standard-error.asp" target = "_blank">https://www.investopedia.com/terms/s/standard-error.asp</a>