Heights of men are normally distributed with a mean of 75 in and a sta

danrussekme 2021-11-21 Answered
Heights of men are normally distributed with a mean of 75 in and a standard deviation of 3.5 in.
a) Find the probability of randomly selecting a man with a height of less than 72 in.
b) Find the probability of randomly selecting a man with a height of greater than 74 in.
c) Find the probability of randomly selecting a man with a height that is between 71 in. and 77 in.

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Answered 2021-11-22 Author has 411 answers

Step 1
NOTE: As per the guidelines, we are supposed to solve the first 3 sub parts only. I have rounded off my values to 4 decimal places.
a) Given:
Mean \(\displaystyle={75}\)
Standard deviation \(\displaystyle={3.5}\)
Data point \(\displaystyle={72}\)
The formula and calculations for the z-score is:
\(\displaystyle{z}=\) Data point-Mean/Standard deviation
\(\displaystyle={72}-{\frac{{{75}}}{{{3.5}}}}\)
\(\displaystyle=-{\frac{{{3}}}{{{3.5}}}}\)
\(\displaystyle=-{0.86}\)
Observe \(\displaystyle{z}=-{0.86}\) in the standard normal table to obtain the required probability.
The answer is 0.1949
Thus, the probability of randomly selecting a man with a height of less than 72 in is 0.1949.
Step 2
b) Given:
Mean \(\displaystyle={75}\)
Standard deviation \(\displaystyle={3.5}\)
Data point \(\displaystyle={74}\)
The formula and calculations for the z-score is:
\(\displaystyle{z}=\) Data point-Mean/Standard deviation
\(\displaystyle={74}-{\frac{{{75}}}{{{3.5}}}}\)
\(\displaystyle=-{\frac{{{1}}}{{{3.5}}}}\)
\(\displaystyle=-{0.29}\)
Observe \(\displaystyle{z}=-{0.29}\) in the standard normal table and subtract the result from 1 to obtain the required probability.
The answer is 0.6141.
Thus, the probability of randomly selecting a man with a height of greater than 74 in is 0.6141.
Step 3
c) Given:
Mean \(\displaystyle={75}\)
Standard deviation \(\displaystyle={3.5}\)
Data point \(\displaystyle={71}\) and 77
The formula and calculations for the first z-score is:
\(\displaystyle{z}_{{{1}}}=\) Data point-Mean/Standard deviation
\(\displaystyle={71}-{\frac{{{75}}}{{{3.5}}}}\)
\(\displaystyle=-{\frac{{{4}}}{{{3.5}}}}\)
\(\displaystyle=-{1.14}\)
The formula and calculations for the first z-score is:
\(\displaystyle{z}_{{{2}}}=\) Data point-Mean/Standard deviation
\(\displaystyle={77}-{\frac{{{75}}}{{{3.5}}}}\)
\(\displaystyle={\frac{{{2}}}{{{3.5}}}}\)
\(\displaystyle={0.57}\)
The required probability is calculated as follows:
\(\displaystyle{P}{\left({71}{<}{X}{<}{77}\right)}={P}{\left(-{1.14}{<}{Z}{<}{0.57}\right)}\)
\(\displaystyle={P}{\left({Z}{<}{0.57}\right)}-{P}{\left({Z}{<}-{1.14}\right)}\)
\(\displaystyle={0.7157}-{0.1271}\)
\(\displaystyle={0.5886}\)
Thus, the probability of randomly selecting a man with a height that is between 71 in. and 77 in is 0.5886.

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