SOLUTION: A manufacturer knows that their items have a normally distributed length, with a mean of 19 inches, and standard deviation of 6.3 inches. If 25 items are chosen at random, what

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Question 1184551: A manufacturer knows that their items have a normally distributed length,
with a mean of 19 inches, and standard deviation of 6.3 inches.
If 25 items are chosen at random, what is the probability that their mean
length is less than 22 inches? (Give answer to 4 decimal places.)
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
A manufacturer knows that their items have a normally distributed length,
with a mean of 19 inches, and standard deviation of 6.3 inches.
If 25 items are chosen at random, what is the probability that their mean
length is less than 22 inches? (Give answer to 4 decimal places.) 

The rule is:

You must use the t-distribution when working problems when the population
standard deviation (σ) is not known and the sample size is small (n<30).
General Correct Rule: If σ is not known, then using t-distribution is
correct. If σ is known, then using the normal distribution is correct.

In this problem, σ is known, so we use the normal distribution.  However,
since the sample size is 25 we must divide the standard deviation by √25.

On your TI-84, press ON CLEAR 2ND VARS 3

Make the screen read like this:

     normalcdf
lower:-99999999
upper:22
μ:19
σ:6.3/√(25)      <-- the √ key is 2ND x2
Paste

Use the down arrow key to highlight Paste
Press ENTER

Read this:

normalcdf(-99999999,22,19,6.3/√(25))

Press ENTER
 
Read 0.9913660393

Round to 0.9914

Edwin