Question

The overhead reach distances of adult females are normally distributed with a mean of

197.5 cm

and a standard deviation of

8.9 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 195.70 cm.

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

Answer 1

SOLUTION:

From given data,

The overhead reach distances of adult females are normally distributed with a mean of 197.5 cm and a standard deviation of 8.9 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 195.70 cm.

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

Where,

mean = = 197.5

standard deviation = = 8.9

Z = (X - ) / =  (X -197.5 ) / 8.9

a. Find the probability that an individual distance is greater than 210.00 cm.

P(X > 210.00) = P( (X - ) / > (210.00 -197.5 ) / 8.9)

P(X > 210.00) = P( Z > (210.00 -197.5 ) / 8.9)

P(X > 210.00) = P( Z > 12.5 / 8.9)

P(X > 210.00) = P( Z > 1.40)

P(X > 210.00) = 1-P( Z < 1.40)

P(X > 210.00) = 1-0.91924 (from z table)

P(X > 210.00) = 0.08076

b. Find the probability that the mean for 20  randomly selected distances is greater than 195.70 cm.

Where ,

n = 20

= / sqrt(n) = 8.9 / sqrt(20) = 1.9901004

= = 197.5

Z = ( - ) / () = ( - 197.5) / (1.9901004)

P( > 195.70 ) = P(( - ) / () > (195.70 -197.5 ) / 1.9901004)

P( > 195.70 ) = P( Z > (195.70 -197.5 ) / 1.9901004)

P( > 195.70 ) = P( Z > -1.8 / 1.9901004)

P( > 195.70 ) = P( Z > -0.90)

P( > 195.70 ) = 1-P( Z < -0.90)

P(X > 195.70 ) = 1-0.18406   (from z table)

P(X > 195.70 ) = 0.81594

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

Because the original population has a normal distribution, the distribution of the sample mean is normal for any sample size.