Question

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.

Short answer

Part(a) The average height of $15$-year-old students at this high school is $168$cm.

Part(b) $4.36$percent of $15$-year-old students at this school are taller than $180$cm

Part(c) No, the slope of the sample regression line will not be exactly $4.2$.

Step-by-step solution

Part(a) Step 1 : Given information

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

Part(a) Step 2 : Simplify

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.

Part(b) Step 1 : Given information

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

Part(b) Step 2 : Simplify

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.

Part(c) Step 1 : Given information

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

Part(c) Step 2 : Simplify

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.