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.