Q.4.15 - Elementary Statistics

Question

Height and Score. A random sample of $10$ students was taken from an introductory statistics class. The following data were obtained, where $X$ denotes height, in inches, and $Y$denotes score on the final exam.

a. What sort of value of $r$ would you expect to find for these data? Explain your answer.

b. Compute $r$.

Step-by-Step Solution

Answer

(a) The expected value of $r$ is about $0$

(b) $r=0.0872$

Part (a) Step 1: Given information

Given table is

We have to explain the expected value of $r$

Part (a) Step 2: Explanation

About $0$ since, in general, it is impossible to establish if the shortest student in a class of students scores worse or higher than the other students. As a result, if$X$ decreases, $Y$ cannot drop or rise.

Part(b) Step1: Given Information

The given table is 

We have to compute $r$

Part (b) Step 2: Explanation

The formula of linear correlation coefficient is 

$r=\frac{\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n}{\sqrt{\left[\sum x_i^2-{\left(\sum x_i\right)}^2/n\right]\left[\sum y_i^2-{\left(\sum y_i\right)}^2/n\right]}}$

As shown in the table below, the appropriate value can be determined.

therefore,

$$\begin{gathered} r=\frac{49596-\left(668\right)\left(742\right)/10}{\sqrt{\left[44700-(668{)}^2/10\right]\left[56622-(742{)}^2/10\right]}} \\ =0.0872 \end{gathered}$$