SOLUTION: A manufacturer produces a commodity where the length of the commodity has approximately normal distribution with a mean of 7.8 inches and standard deviation of 2.6 inches. If a sam

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Question 1203897: A manufacturer produces a commodity where the length of the commodity has approximately normal distribution with a mean of 7.8 inches and standard deviation of 2.6 inches. If a sample of 49 items are chosen at random, what is the probability the sample's mean length is greater than 8.8 inches? Round answer to four decimal places.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
population mean = 7.8
population standard deviation = 2.6
sample size = 49
sample mean = 8.8

standard error = population standard deviation / sqrt(sample size) = 2.6 / sqrt(49) = .37143.

use z = (x-m)/s formula.
z = z-score
x is sample mean
m = population mean
s = standard error

formula becomes z = (8.8-7.8)/.37143 = 2.692297.

area to the right of that z-score = .003548.
round to .0035.
there is a 3.5% probability that the z-score wil be more than 8.8.

here's what the results look like using the calculator at https://davidmlane.com/hyperstat/z_table.html

the calculator can do it from the z-score or the raw score.
if z-score, mean = 0 and standard deviation = 1.
if raw score, mean = 7.8 and standard deviation = standard error of .37143.