Q. 4.150 - Elementary Statistics

Question

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

The answers to both the subparts have been given below.

Step 1. Given Information

A table has been given:

Step 2. The answer for (a)

As a random sample of $10$ was taken from an introductory statistics class. The following data were obtained, where $x$ denoted height in inches, and $y$ denoted score on the final exam.

The variables height and score are independent of each other. So, we expect the value of $r$ to be $0$ or very close to $0$.

Step 3. The answer for (b)

The computation for the linear correlation coefficient is as follows:

The formula for the linear correlation coefficient is:

$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{n\sum x^2-{(\sum x)}^2}\sqrt{n\sum y^2-{(\sum y)}^2}}$
To find the linear correlation coefficient, substitute the known values in the formula
$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{n\sum x^2-{(\sum x)}^2}\sqrt{n\sum y^2-{(\sum y)}^2}}$
$=\frac{10\left(49596\right)-\left(668\right)\left(742\right)}{\sqrt{10\left(44700\right)-{\left(668\right)}^2}\sqrt{10\left(56622\right)-{\left(742\right)}^2}}$

$$\begin{gathered} =0.087217 \\ \approx 0.087 \end{gathered}$$

So, the correlation coefficient is $r=0.087$