Question

The heights of a female population follow a normal distribution with a mean of 48 inches and a standard deviation of 6 inches. If a random sample of 16 subjects were taken, what is the probability that the average height of the sample is higher than 50 inches?

Answer 1

If X follows a Normal distribution with mean and standard deviation , then sampling distribution of sample mean: follows a normal distribution with mean and standard deviation

Given,

X: heights of a female population

X follow a normal distribution with a mean of 48 inches and a standard deviation of 6 inches

Therefore  average height of the sample of 16(n=16) subjects ( ) follows a normal distribution with mean 48 inches and standard deviation ( = 6/4 = 1.5) of 1.5

Probability that the average height of the sample is higher than 50 inches = P( > 50)

P( > 50) = 1 - P( 50)

Z-score for 50 = (50-mean)/Standard deviation = (50-48)/1.5 = 2/1.5 = 1.33

From Standard normal tables, P(Z 1.33) = 0.9082

P( 50) = P(Z 1.33) = 0.9082

P( > 50) = 1 - P( 50) = 1-0.9082 = 0.0918

Probability that the average height of the sample is higher than 50 inches = P( > 50) = 0.0918

Probability that the average height of the sample is higher than 50 inches = 0.0918