Chapter 5, Problem 1TP

1.

To determine

The Pearson product-moment correlation coefficient.

Answer to Problem 1TP

The correlation coefficient between number correct (out of a possible 20) and Attitude (out of a possible 100) is 0.59.

Explanation of Solution

The given information represents the data set as follows:

Number correct (out of a possible 20)	Attitude (out of a possible 100)
17	94
13	73
12	59
15	80
16	93
14	85
16	66
16	79
18	77
19	91

Calculation:

The formula for calculating Pearson product moment is given by;

${\text{r}}_{\text{XY}}\frac{n{\displaystyle \sum \text{XY}-{\displaystyle \sum \text{X}{\displaystyle \sum \text{Y}}}}}{\sqrt{\left[n{\displaystyle \sum {\text{X}}^2-{\left({\displaystyle \sum \text{X}}\right)}^2}\right]\left[n{{\displaystyle \sum \text{Y}}}^2-{\left({\displaystyle \sum \text{Y}}\right)}^2\right]}}$ (1)

• $r_{\text{XY}}$ is the correlation coefficient between X and Y.
• n is the size of the sample.
• X is the number correct (out of a possible 20) on the X variable.
• Y is the Attitude (out of a possible 100) on the Y variable.

Number correct (X)	Attitude (Y)	${\text{X}}^{\text{2}}$	${\text{Y}}^{\text{2}}$	$\text{XY}$
17	94	289	8836	1598
13	73	169	5329	949
12	59	144	3481	708
15	80	225	6400	1200
16	93	256	8649	1488
14	85	196	7225	1190
16	66	256	4356	1056
16	79	256	6241	1264
18	77	324	5929	1386
19	91	361	8281	1729
156	797	2476	64727	12568

Substitute 156 for ${\displaystyle \sum \text{X}}$, 797 for ${\displaystyle \sum \text{Y}}$, 2476 for ${\displaystyle \sum {\text{X}}^2}$, 6472 for ${\displaystyle \sum {\text{Y}}^2}$, 12568 for ${\displaystyle \sum \text{XY}}$ and 10 for n in the above equation (1).

$\begin{array}{c}{\text{r}}_{\text{XY}}=\frac{10\times 12568-156\times 797}{\sqrt{\left[10\times 2476-{\left(156\right)}^2\right]\left[10\times 6472-{\left(797\right)}^2\right]}}\\ =\frac{1348}{\sqrt{\left[24760-24336\right]}\left[647270-635209\right]}\\ =\frac{1348}{\sqrt{424\times 12061}}\\ =0.59\end{array}$

The correlation between number correct and attitude is positive because the value of correlation coefficient is (0.59) lies in between 0 to 1.

2.

To determine

The scatter plot for the 10 pairs of correct and attitude; whether direct or indirect correlation would be predicted or not based on the scatterplot.

Answer to Problem 1TP

There is a direct relationship between number of correct and attitude because, if the value of correct increases then the value of attitude will also increase.

Explanation of Solution

Calculation:

Take number correct on the X-axis and attitude on the Y-axis to sketch the scatter plot.

The scattered plot is obtained as:

Fig (1)

The above scatter plot represents the direct relationship between number of correct and attitude because, if the value of number correct increases then the value of attitude will also increase.