Predicting height Using the health records of every 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 ${\mu }_y=105+4.2x$with $\sigma =7$cm.

a. According to the population regression line, what is the average height of $15$-year-old students at this high school?

b. About what percent of $15$-year-old students at this school are taller than $180$cm?

c. If the nurse used a random sample of $50$students from the school to calculate the regression line instead of using all the students, would the slope of the sample regression line be exactly $4.2$? Explain your answer.

W need to find the average height of $15$-year-old students at this high school.

According to the population regression line,

$$\begin{gathered} {\mu }_y=105+4.2x \\ \sigma =7 \end{gathered}$$

For average height,

$$\begin{gathered} {\mu }_y=105+4.2x \\ =105+4.2(1.5) \\ =105+63 \\ =168 \end{gathered}$$

Hence, the average height of $15$-year-old students at this high school is $168$ cm.

We need to find percentage of $15$-year-old students at this school who are taller than $180$ cm.

According to formula z-score is :

$z=\frac{x-\mu }{\sigma }$

Average mean is

$$\begin{gathered} {\mu }_y=105+4.2x \\ =105+4.2(1.5) \\ =105+63 \\ =168 \end{gathered}$$

and $\sigma =7$,

Now, substituting the values

$z=\frac{x-\mu }{\sigma }=\frac{180-168}{7}=1.71$

Probability for required case :

$$\begin{gathered} P(X>180)=P(Z>180) \\ =1-P(Z<180) \\ =1-0.9564 \\ =4.36\% \end{gathered}$$

Hence, $4.36\%$students at this school are taller than $180$ cm.

We need to check would the slope of the sample regression line be exactly $4.2$ .

No, the slope of the sample regression line will not be exactly $4.2$but it will be near to $4.2$

Because there will be some variability in the random sample that had been taken by the nurse to school to calculate the regression line instead of using every student of school.