Question 1120034
the 95% confidence interval requires a critical z-score of plus or minus 1.959963986.


the z-score formula is:


z = (x - m) / s


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


your raw mean is 260 hours.
your population standard deviation is 75 hours.
your sample size is 36.


formula for standard error is:


standard error = standard deviation / square root of size.


in your problem, you get:


s = 75 / sqrt(36) = 75 / 6 = 12.5.


use the critical z-scores and the standard error and the mean to find the critical raw scores.


formula is z = (x - m) / s


z = plus or minus 1.959963986.


s = 12.5


on the low side, the z-score formula becomes:


-1.959963986 = (x - 260) / 12.5


solve for x to get x = 235.5004502


on the high side, the z-score formula becomes:


1.959963986 = (x - 260) / 12.5


solve for x to get x = 284.4995498.


that's your 95% confidence interval.


following is a picture of the 95% confidence interval using z-scores.


<img src = "http://theo.x10hosting.com/2018/071815.jpg" alt="$$$" >


following is a picture of the 95% confidence interval using raw scores.


<img src = "http://theo.x10hosting.com/2018/071816.jpg" alt="$$$" >


in both, the area under the normal distribution curve is shown as .95.


that's the same as 95%.


your confidence interval is the shaded area on the graph.


the critical z-score was found using the inverse norm function of the TI-84 Plus calculator.