In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 67.7 inches and a standard deviation of 4.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below. (a) Find the probability that a study participant has a height that is less than 67 inches. The probability that the study participant selected at random is less than 67 inches tall is nothing. (Round to four decimal places as needed.) (b) Find the probability that a study participant has a height that is between 67 and 72 inches. The probability that the study participant selected at random is between 67 and 72 inches tall is nothing. (Round to four decimal places as needed.) (c) Find the probability that a study participant has a height that is more than 72 inches. The probability that the study participant selected at random is more than 72 inches tall is nothing. (Round to four decimal places as needed.) (d) Identify any unusual events. Explain your reasoning. Choose the correct answer below. A. There are no unusual events because all the probabilities are greater than 0.05. B. The event in part (a) is unusual because its probability is less than 0.05. C. The events in parts (a) and (c) are unusual because its probabilities are less than 0.05.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of

67.7

inches and a standard deviation of

4.0

inches. A study participant is randomly selected. Complete parts (a) through (d) below.

(a) Find the probability that a study participant has a height that is less than

inches.

The probability that the study participant selected at random is less than

inches tall is

nothing.

(Round to four decimal places as needed.)

(b) Find the probability that a study participant has a height that is between

and

inches.

The probability that the study participant selected at random is between

and

inches tall is

nothing.

(Round to four decimal places as needed.)

inches.

The probability that the study participant selected at random is more than

inches tall is

nothing.

(Round to four decimal places as needed.)

(d) Identify any unusual events. Explain your reasoning. Choose the correct answer below.

There are no unusual events because all the probabilities are greater

than 0.05.

The event in part

(a)

is unusual because its probability is less than 0.05.

The events in parts

(a) and (c)

are unusual because its probabilities are less than 0.05.

Expert Solution

Step 1

Let X denote the heights of the group of men.

It is given that,

$X ~ N (67.7, 4^{2})$

a. To find P[X<67]

It is calculated using CLT.

$P [X < 67] = P [\frac{X - 67.7}{4} < \frac{67 - 67.7}{4}] = P [Z < - 0.175] = ϕ (- 0.175) T h e a b o v e v a l u e i s f o u n d u \sin g d i s t r i b u t i o n t a b l e s . P [X < 67] = 0.42858$

Hence, the required probability is 0.4286. (rounded off to 4 decimal places)

b. To calculate P[67<X<72]

It can be found out using CLT.

$P [67 < X < 72] = P [\frac{67 - 67.7}{4} < \frac{X - 67.7}{4} < \frac{72 - 67.7}{4}] = P [- 0.175 < Z < 1.075] = ϕ (1.075) - ϕ (- 0.175) t h e v a l u e s a r e f o u n d u \sin g s \tan d a r d n o r m a l t a b l e s . T h e r e f o r e, P [67 < X < 72] = 0.85993 - 0.42858 = 0.43135$

Hence, the required probability is 0.4314. (rounded off to 4 decimal places)

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