Question

The overhead reach distances of adult females are normally distributed with a mean of 197.5 cm and a standard deviation of 7.8 cm .

a. Find the probability that an individual distance is greater than 207.50 cm.
b. Find the probability that the mean for 15 randomly selected distances is greater than 196.20 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 _________
(Round to four decimal places as needed.)

b. The probability is _________
(Round to four decimal places as needed.)

c. Choose the correct answer below.

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

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

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

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

Answer 1

Solution :

Given that ,

mean = = 197.5

standard deviation = = 7.8

(a)

P(x > 207.5) = 1 - P(x < 207.5)

= 1 - P((x - ) / < (207.5 - 197.5) / 7.8)

= 1 - P(z < 1.282)

= 1 - 0.90

= 0.10

Probability = 0.1000

(b)

n = 15

= 197.5 and

= / n = 7.8 / 15

P( > 196.20) = 1 - P( < 196.20)

= 1 - P(( - ) / < (196.20 - 197.5) / 7.8 / 15 )

= 1 - P(z < -0.65)

= 1 - 0.2578

= 0.7422

Probability = 0.7422

(c)

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