Question

n a survey of a group of​ men, the heights in the​ 20-29 age group were normally​ distributed, with a mean of 67.8 inches and a standard deviation of 3.0 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. The events in parts left parenthesis a right parenthesis and left parenthesis c right parenthesis are unusual because its probabilities are less than 0.05. B. The events in parts left parenthesis a right parenthesis comma left parenthesis b right parenthesis comma and left parenthesis c right parenthesis are unusual because all of their probabilities are  less than 0.05. C. There are no unusual events because all the probabilities are greater than 0.05. D. The event in part left parenthesis a right parenthesis is unusual because its probability is less than 0.05.

Answer 1

Answer to the question)

Given:

Mean ( M ) = 67.8

Standard deviation ( s) = 3

.

Formula of Z score =

Z = (x - M) / s

Where x denotes the value for Z score is calculated

.

Part a)

To find probability less than 67 inches P(X < 67)

P(x < 67) = P(z < (67 -67.8)/3) = P(z < -0.27) = 0.3936

P(x<67) = 0.3936

[we get the probability value from the Z table , as shown below]

.

.

Part b)

P(67 < x < 72) = P(x < 72) - P(x < 67)

from part (a) we know that P(x < 67) = 0.3936

We find P(x < 72) = P(z (72 -67.8)/3) = P(z<1.4) = 0.9192 [ refer to Z table]

.

.

Thus P(67 < x < 72) = 0.9192 - 0.3936 = 0.5256

Thus P(67 < x < 72) = 0.5256

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Part c)

P(x > 72) = 1 - P(x < 72)

[since the total area under the normal curve is 1]

.

We know that P(x<72) = 0.9192

Thus,

P(x > 72) = 1 - 0.9192

P(x > 72) = 0.0808

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Part d)

The above three events have:

P(x < 67) = 0.3936

P(67 < x < 72) = 0.5256

P(x > 72) = 0.0808

All these probabilities are greater than 0.05, hence none of these events is unusual.

Thus answer choice ( C ) is correct. There are no unusual events because all the probabilities are greater than 0.05