Question

The overhead reach distances of adult females are normally distributed with a mean of 200 cm and a standard deviation of 7.8 cm.

a. Find the probability that an individual distance is greater than 212.50cm.

b. Find the probability that the mean for 15 randomly selected distances is greater than 198.50 cm

c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

Answer 1

Answer)

As the data is normally distributed we can use standard normal z table to estimate the answers

Z = (x-mean)/s.d

Given mean = 200

S.d = 7.8

A)

P(x>212.5)

Z = (212.5-200)/7.8 = 1.6

From z table, P(z>1.6) = 0.0548

B)

Z = (x - mean)/(s.d/√n) for a sample

So,

Z = (198.5-200)/(7.8/√15) = -0.74

From z table, P(z>-0.74) = 0.7704

C)

Yes, as according to the central limit theorem any sample from the normally distributed population is normally distributed itself.