Question 1206525
mean is 257.2 cm.
standard deviation is 1.5 cm
sample size is 12.


standard error = population standard deviation / sqrt(sample size) = 1.5 / sqrt(12) = .433013.


you want to know the probability that the mean of the sample will be between 257.7 and 258.1 cm.


it appears the problem is looking for z-score results, so we'll go with that.


z-score formula is z = (x - m) / s


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


standard error = standard deviation / sqrt(sample size) = 1.5 / sqrt(12) = .433013 rounded to 6 decimal places.


you want to find the z-scores for sample means of 257.7 and 258.1.


z-score formula for sample mean of 257.7 is z = (257.7 - 257.2) / .433013 = 1.1547.


z-score formula for sample mean of 258.1 is z = (258.1 - 257.2) / .433013 = 2.0785.


you can use a z-score calculator, such as the one at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "_blank">https://davidmlane.com/hyperstat/z_table.html</a>


use of that calculator tells you that the area between those 2 z-score is equal to .1053.


that's the probability that a sample of size 12 will have a mean length of between 257.7 and 258.1 centimeters.


here's what that looks like.


<img src = "http://theo.x10hosting.com/2024/031803.jpg">