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

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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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.
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.
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.