Engineers must consider the breadths of male heads when desi

Engineers must consider the breadths of male heads when designing motorcycle helmets. Men have head breadths that are normally distributed with a mean of 6.5 inches and a standard deviation of 1.1 inches.

(a) If one male is randomly selected, find the probability that his head breadth is less than 5.9 inches.

(b) If a random sample of 64 males is selected, find the probability that the mean of their head breadths is less than 6.1 inches.

(c) Is there a difference in the two answers?

Solution

Engineers must consider the breadths of male heads when designing motorcycle helmets. Men have head breadths that are normally distributed with a mean of 6.5 inches and a standard deviation of 1.1 inches.

(a) If one male is randomly selected, find the probability that his head breadth is less than 5.9 inches.

Z value for 5.9, z=(5.9-6.5)/1.1 =-0.55

P( x < 5.9) = P( z< -0.55) = 0.2912

(b) If a random sample of 64 males is selected, find the probability that the mean of their head breadths is less than 6.1 inches.

Standard error = sd/sqrt(n) =1.1/sqrt(64) =0.1375

Z value for 6.1, z=(6.1-6.5)/0.1375 =-2.91

P( x < 6.1) = P( z< -2.91)

= 0.0018

(c) Is there a difference in the two answers? Explain your answer.

Yes, there is a difference in the two answers. In the first, individual value is used. In the second, mean value is used and for this we calculated standrd error of the mean.