In: Math
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?
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