Normal distribution

Question

Given a normal distribution with 𝜇 = 12 and 𝜎 = 10. If a random sample of size n = 70 is drawn, find P(12 ≤ x ≤ 14).

Round to three decimal places.

Answer 1

Assuming X is the mean of the sample of size 70, then X also follows a normal distribution with the same mean=12 but with a new standard deviation which is the old standard deviation but divided by sqrt(n) where n is the sample size.

Thus the new standard deviation for the sampling distribution is: 10/sqrt(70).

Then to find probabilities for a normal distribution, you need to find z-score using the z-score formula but with your new standard deviation SD=10/sqrt(70).

z=(x-mean)/SD

For x=12, converting that to z=(12-12)/(10/sqrt(70))=0

For x=14, converting that to z=(14-12)/(10/sqrt(70))=1.6733

So then you want to find the area between z=0 and z=1.6733 using a z-score table.

The area to the left of z=0 under the standard normal distribution is 0.5.

The area to the left of z=1.6733 is 0.9529

Thus the area BETWEEN z=0 and z=1.6733 is 0.9529-0.5=0.4529.

Rounded to three decimal places, this is 0.453