Question 1186035
men of the population is 829
standard deviation of the population is 50.


critical z-score for admission to be in the top 1 percent is equal to 2.326348 rounded to 6 decimal places.


z-score formula is:


z = (x - m) / s


z is the z-score
x is the individual score
m is the mean
s is the standard deviation


the critical z-score formula becomes:


2.326348 = (x - 829) / 50.


solve for x to get:


x = 2.326348 * 50 + 829 = 945.3174.


sample size = 42.


when you're dealing with the mean of a sample, rather than an individual score, you need to use the standard error.


standard error = standard deviation / square root of sample size = 50 / sqrt(42) = 7.715167 rounded to 6 decimal places.


z = (x - m) / s


z is the z-score
x is the sample mean
m is the population mean
s is the standard error


formula becomes:


z = (849 - 829) / 7.715167 = 2.5923 rounded to 4 decimal places.


area to the right of that z-score is equal to .00477 rounded to 5 decimal places.


that's the probability that the group average is not less than 849.


i used the following calculator to confirm these results are accurate, assuming i calculated the standard error correctly.


<a href = "https://www.statskingdom.com/normal.html" target = "_blank">https://www.statskingdom.com/normal.html</a>


the results from the use of that calculator are shown below.


minimum score for top 1%:


<img src = "http://theo.x10hosting.com/2021/101401.jpg" >


probability of sample mean not less than 849.


<img src = "http://theo.x10hosting.com/2021/101402.jpg" >