# Using the health records of ever student at a high school, the school nurse created a scatterplot relating y = height (in centimeters) to x = age (in years).

Using the health records of ever student at a high school, the school nurse created a scatterplot relating y = height (in centimeters) to x = age (in years).
After verifying that the conditions for the regression model were met, the nurse calculated the equation of the population regression line to be .
About what percent of 15-year-old students at this school are taller than 180 cm?

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Step 1
Given:
${\mu }_{y}=105+4.2x$ (Equation population regression line)
$\sigma =7$
The average height of 15-year-old students at this high school according to the population regression line can be found by replacing x in the regression line equation by 15 and evaluating.
${\mu }_{y}=105+4.2\left(15\right)=105+63=168$
Thus the mean is 168 and the standard deviation is 7.
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.
$z=\frac{x-\mu }{\sigma }=\frac{180-168}{7}\approx 1.71$
Determine the corresponding probability using the normal probability table in the appendix. $P\left(Z<1.71\right)$ is given in the row starting with 1.7 and in the column starting with .01 of the standard normal probability table in the appendix.
$P\left(X>180\right)=P\left(Z>1.71\right)$
$=1-P\left(Z<1.71\right)$
$=1-0.9564$
$=0.0436$
$=4.36\mathrm{%}$
Thus about 4.36% of the 15-year-old students at this school are expected to be taller than 180 cm.
Result: 4.36%