Question

The overhead reach distances of adult females are normally distributed with a mean of 205.5 cm and a standard deviation of 8 cm. a. Find the probability that an individual distance is greater than 218.90 cm. b. Find the probability that the mean for 25 randomly selected distances is greater than 203.30 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 nothing. ​(Round to four decimal places as​ needed.)

b. The probability is nothing. ​(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 original population has a normal distribution.

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

Answer 1

Mean, = 205.5 cm

Standard deviation, = 8 cm

a) P(X < A) = P(Z < (A - )/)

P(individual distance is greater than 218.90 cm) = P(X > 218.90)

= 1 - P(X < 218.90)

= 1 - P(Z < (218.90 - 205.5)/8)

= 1 - P(Z < 1.675)

= 1 - 0.9530

= 0.0470

b) Sample size, n = 25

For sampling distribution of mean,

P( < A) = P(Z < (A - )/)

= = 205.5 cm

=

=

= 1.6

P( > 203.3 cm) = 1 - P( < 203.3)

= 1 - P(Z < (203.3 - 205.5)/1.6)

= 1 - P(Z < -1.375)

= 1 - 0.0846

= 0.9154

c) C. The normal distribution can be used because the original population has a normal distribution.