Chapter 12.1, Problem 1E

(a)

To determine

To find: the average height of 15-year old students.

Answer to Problem 1E

168 centimetres

Explanation of Solution

Given:

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

Calculation:

For the average height = 15 years

  $\begin{array}{c}{\mu }_y=105+4.2x\\ =105+4.2\left(15\right)\\ =105+63\\ =168\end{array}$

Therefore the average height as per the population regression line is 168 centimetres.

(b)

To determine

To find: the percent of 15-year old students are taller than 180 centimetres.

Answer to Problem 1E

4.36%

Explanation of Solution

Given:

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

Formula used:

  $z=\frac{x-\mu }{\sigma }$

Calculation:

Average mean

  $\begin{array}{c}{\mu }_y=105+4.2x\\ =105+4.2\left(15\right)\\ =105+63\\ =168\end{array}$

  $\sigma =7$

Z score is

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{180-168}{7}\\ =1.71\end{array}$

Corresponding probability for the taller than 180 centimetre

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

Therefore about 4.36 percent of the 15-year old students are expected to be taller than 180 centimetres.

(c)

To determine

To find: that the slope of the sample regression line be exactly 4.2, yes or not.

Answer to Problem 1E

No

Explanation of Solution

No, the reason is that the slope of the population regression line is 4.2 and it is predicted that the slope of the regression line of sample is very near to 4.2 but it is not exactly so there would be some sampling variability in a sample.