Normal Distribution

Question

A manufacturer knows that their items have a normally distributed length, with a mean of 17.3 inches, and standard deviation of 2.1 inches.

If 12 items are chosen at random, what is the probability that their mean length is less than 16.1 inches?

P(¯x<16.1) = 

 

Answer 1

We need to take account of the sample size to make an adjustment for the population standard deviation, so s, the sample deviation=2.1/√12=0.6062, where 12 is the sample size.

The difference between the sample mean and the population mean is 17.3-16.1=1.2 inches below the population mean. This is 1.2/0.6062=1.9795 standard deviations below the mean and is the Z value of -1.9795. This corresponds to a probability of 0.02388 in the normal distribution.

Therefore P(X<16.1)=0.0239 approx (2.39%).