Chapter 15.4, Problem 9WE

To determine

To calculate: Theestimate of the student’s score in the second test.

Answer to Problem 9WE

The estimate of the student’s score in the second testis $71$ .

Explanation of Solution

Given information:

100 students of junior class gave two test and correlation coefficient of two tests is 0.70. The mean of first test was 125 and mean of second test was 80. Standard deviation of first test is 20 and for second test is 10.

Formula used:

The correlation coefficient r is denoted as $r=\frac{M_{xy}-M_x\cdot M_y}{{\sigma }_x\cdot {\sigma }_y}$ , where $M_{xy}$ is the mean of the product of values of x and y , $M_x$ is the mean of x -values, $M_y$ is the mean of y -values, ${\sigma }_x$ is standard deviation of x -values, ${\sigma }_y$ is the standard deviation of y -values.

The regression line that relates x and y , contains the point $\left(M_x,M_y\right)$ has slope $r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)$ and the equation is $y-M_y=r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)\left(x-M_x\right)$ .

Calculation:

Consider the provided information that 100 students of junior class gave two test and correlation coefficient of two tests is 0.70. The mean of first test was 125 and mean of second test was 80. Standard deviation of first test is 20 and for second test is 10.

Let x denote the score of test 1 and y denote the score of second test.

Here, $r=0.70,M_x=125,M_y=80,{\sigma }_x=20\text{ and }{\sigma }_y=10$ .

Next, recall that the regression line that relates x and y , contains the point $\left(M_x,M_y\right)$ has slope $r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)$ and the equation is $y-M_y=r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)\left(x-M_x\right)$ .

Since, student obtained 100 marks in first test so $x=100$ .

Substitute the provided values in the expression provided above,

  $\begin{array}{c}y-80=0.70\left(\frac{10}{20}\right)\left(100-125\right)\\ y-80=\frac{7}{20}\left(-25\right)\\ y=-\frac{35}{4}+80\\ y=71.25\end{array}$

Therefore, approximately the value of $y\approx 71$ .

Thus, the estimate of the student’s score in the second test is $71$ .