Question

3. The heights of female students attending a sixth form college have a mean of 168 cm and standard deviation of 4.5 cm. The heights can be modeled by a normal distribution: a. Find the probability that the height of a randomly selected female student attending this college is less than 172.5 cm? b. Find the probability that the mean height of a random sample of 11 female students from this college exceeds 172 cm?

Answer 1

Solution:

Given: The heights of female students attending a sixth form college follows a Normal distribution with a mean of 168 cm and a standard deviation of 4.5 cm.

That is:  

Part a) Find the probability that the height of a randomly selected female student attending this college is less than 172.5 cm.

That is find:

P( X < 172.5 ) =..........?

Find z score for x = 172.5

Thus we get:

P( X < 172.5 ) =P( Z< 1.00)

Look in z table for z= 1.0 and 0.00 and find corresponding area.

P(Z < 1.00) = 0.8413

Thus

P( X < 172.5 ) =P( Z< 1.00)

P( X < 172.5 ) =0.8413

Thus the probability that the height of a randomly selected female student attending this college is less than 172.5 cm is 0.8413

Part b) Find the probability that the mean height of a random sample of 11 female students from this college exceeds 172 cm

Sample size = n = 11

That is find :

Use following z score formula for sample mean.

Thus we get:

Look in z table for z = 2.9 and 0.05 and find corresponding area.

P( Z < 2.95 ) = 0.9984

Thus

Thus the probability that the mean height of a random sample of 11 female students from this college exceeds 172 cm is 0.0016