Question

Use the information about the overhead reach distances of adult females: u= 205.5, u= 8.6 cm, and overhead reach distances are normally distributed.  The overhead reach distances are used in planning assembly work stations.

If one adult female is randomly selected, find the probability that her overhead reach is less than 196.9 cm.
If one adult female is randomly selected, find the probability that her overhead reach is between 179.7 and 231.3 cm.
If 25 adult females are randomly selected, find the probability that they have a mean overhead reach greater than 203.0 cm.
Why can the normal distribution be used in part (c), even though the sample size does not exceed 30?

Answer 1

Note: u=8.6 should be =8.6 to solve the problem this must be a typing error here.

1. We want P(X<196.9)

Using standard normal approximation,

P(Z< 196.9 - 205.5/8.6) = P(Z<-1) = 1 - P(Z<1) = 1- 0.84134 = 0.15866

2. P(179.7<X<231.3) = P(-3<Z<3) = P(Z<3) - P(Z<-3) = 2P(Z<3) - 1 =

2*0.99865 - 1 = 0.9973

3. Here X follows a N(205.5, 8.6/√25) distribution

So, P(X>203) = P(Z> 203- 205.5/1.72) = P(Z>-1.45) = P(Z<1.45) = 0.92647

we can use the normal distribution here even though n=25 is less than 30, because the population SD of 8.6 is known.