Given:
(Equation population regression line):
\(\displaystyle\mu_{{{y}}}={105}\ +\ {4.2}{x}\)

\(\displaystyle\sigma={7}\) The average height of 15-year-old students at this high scool according to the population regression line can be found by replacing x in the regression line equation by 15 and evaluating. \(\displaystyle\mu_{{{y}}}={105}\ +\ {4.2}{\left({15}\right)}={105}\ +\ {63}={168}\) Thus the mean is 168 and the standard deviation is7. Since the conditions are met, the response y varies according to a Normal distribution. The z-score is the value decreased by the mean, divided by the standard deviation. \(\displaystyle{z}=\ {\frac{{{x}\ -\ \mu}}{{\sigma}}}=\ {\frac{{{180}\ -\ {168}}}{{{7}}}}\ \approx\ {1.71}\) Determine the corresponding probability using the normal probability table in the appendix. \(\displaystyle{P}{\left({Z}\ {<}\ {1.71}\right)}\)</span> is given in the row starding with 1.7 and in the column starting with .01 of the standard normal probability table in the appendix. \(\displaystyle{P}{\left({X}\ {>}\ {180}\right)}={P}{\left({Z}\ {>}\ {1.71}\right)}\)

\(\displaystyle={1}\ -\ {P}{\left({Z}\ {<}\ {1.71}\right)}\)</span>

\(\displaystyle={1}\ -\ {0.9564}\)

\(\displaystyle={0.0436}\)

\(\displaystyle={4.36}\%\) Thus about \(\displaystyle{4.36}\%\) of the 15-year-old students at this scool are expected to be taller than 180 cm.

\(\displaystyle\sigma={7}\) The average height of 15-year-old students at this high scool according to the population regression line can be found by replacing x in the regression line equation by 15 and evaluating. \(\displaystyle\mu_{{{y}}}={105}\ +\ {4.2}{\left({15}\right)}={105}\ +\ {63}={168}\) Thus the mean is 168 and the standard deviation is7. Since the conditions are met, the response y varies according to a Normal distribution. The z-score is the value decreased by the mean, divided by the standard deviation. \(\displaystyle{z}=\ {\frac{{{x}\ -\ \mu}}{{\sigma}}}=\ {\frac{{{180}\ -\ {168}}}{{{7}}}}\ \approx\ {1.71}\) Determine the corresponding probability using the normal probability table in the appendix. \(\displaystyle{P}{\left({Z}\ {<}\ {1.71}\right)}\)</span> is given in the row starding with 1.7 and in the column starting with .01 of the standard normal probability table in the appendix. \(\displaystyle{P}{\left({X}\ {>}\ {180}\right)}={P}{\left({Z}\ {>}\ {1.71}\right)}\)

\(\displaystyle={1}\ -\ {P}{\left({Z}\ {<}\ {1.71}\right)}\)</span>

\(\displaystyle={1}\ -\ {0.9564}\)

\(\displaystyle={0.0436}\)

\(\displaystyle={4.36}\%\) Thus about \(\displaystyle{4.36}\%\) of the 15-year-old students at this scool are expected to be taller than 180 cm.