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 210.00 cm. b. Find the probability that the mean for 20 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?

a. The probability is

b. The probability is

c. Choose the correct answer below.

A.The normal distribution can be used because the mean is large.

B.The normal distribution can be used because the probability is less than 0.5

C.The normal distribution can be used because the finite population correction factor is small.

D.The normal distribution can be used because the original population has a normal distribution.

Answer 1

Given,

= 200 , = 7.8

We convert this to standard normal as

P( X < x) = P (Z < x - / )

A)

P( X > 210) = P( Z > 210 - 200 / 7.8)

= P( Z > 1.2821)

= 1 - P( Z < 1.2821)

= 1 - 0.9001

= 0.0999

b)

Using central limit theorem,

P( < x) = P( Z < x - / / sqrt(n) )

So,

P( > 198.5) = P( Z > 198.5 - 200 / (7.8 / sqrt(20) )

= P (Z > -0.86)

= P( Z < 0.86)

= 0.8051

C)

Normal distribution is used in part B because , the original population has a normal distribution.