Chapter 15, Problem 12CAP

1.

To determine

Identify the correlation coefficient that should be used when studying relation between Activity (active, inactive) and depression (depressed, not depressed).

Answer to Problem 12CAP

Phi-correlation coefficient should be used when studying relation between when studying relation between Activity (active, inactive) and depression (depressed, not depressed).

Explanation of Solution

The given information is that, correlation coefficient between Activity (active, inactive) and depression (depressed, not depressed) is to be studied.

Phi-correlation coefficient:

When the two factors are measured on nominal scale, then the strength and direction of the linear relationship between the two dichotomous factors can be measured by Phi-correlation coefficient. This is denoted by $r_{\phi }$. The formula is,

$r_{\phi }=\frac{ad-bc}{\sqrt{ABCD}}$

The table denoting the notation of Phi-correlation coefficient is,

		Variable X
		$X_1$	$X_2$
Variable Y	$Y_1$	a	b	A
	$Y_2$	c	d	B
		C	D

Table 1

Justification: The factors in the study are Activity (active, inactive) and depression (depressed, not depressed). The factor ‘Activity’ is dichotomous which has two levels ‘active, inactive’ and factor ‘depression’ is dichotomous which has two levels ‘depressed, not depressed’. The phi-correlation coefficient is used to measure the relationship when two factors are dichotomous.

Hence, Phi-correlation coefficient should be used when studying relation between when studying relation between Activity (active, inactive) and depression (depressed, not depressed).

2.

To determine

Identify the correlation coefficient that should be used when studying relation between time spent at school and time spent studying in hours per week.

Answer to Problem 12CAP

Pearson correlation coefficient should be used when studying relation between times spent at school and time spent studying in hours per week.

Explanation of Solution

The given information is that, correlation coefficient between Time spent at school and time spent studying in hours per week is to be studied.

Pearson correlation coefficient:

When the two factors are measured on either interval scale or ratio scale, then the strength and direction of the linear relationship between the two factors can be measured by Pearson correlation coefficient. This is also termed as product-moment correlation coefficient. The formula is,

$r=\frac{S{S}_{XY}}{\sqrt{S{S}_{X}S{S}_Y}}$

In the formula, $S{S}_{XX}$ denotes the sum of squares for factor X and $S{S}_{YY}$ denotes the sum of squares for factor Y.

Justification: The factors in the study are ‘times spent at school’ and ‘time spent studying in hours per week’. These two factors involve times which are measured on ratio scale. The Pearson correlation coefficient is used to measure the relationship when two factors are measured on ratio scale.

Hence, Pearson correlation coefficient that should be used when studying relation between times spent at school and time spent studying in hours per week.

3.

To determine

Identify the correlation coefficient that should be used when studying relation between Veteran (yes, no) and level of patriotism indicated on a rating scale.

Answer to Problem 12CAP

Point-biserial correlation coefficient should be used when studying relation between Veteran (yes, no) and level of patriotism indicated on a rating scale.

Explanation of Solution

The given information is that, correlation coefficient between Veteran (yes, no) and level of patriotism indicated on a rating scale is to be studied.

Point-biserial correlation coefficient:

When the one factor is continuous that is measured on either interval scale or ratio scale and other is dichotomous that is measured on nominal scale, then the strength and direction of the linear relationship between the two factors can be measured by Point-biserial correlation coefficient. It is denoted by $r_{\text{pb}}$. The formula is,

$r_{\text{pb}}=\left(\frac{M_{Y_1}-M_{Y_2}}{s_Y}\right)\left(\sqrt{pq}\right)$

In the formula, p and q are the proportion of scores for the dichotomous factor at each level, $s_Y=\sqrt{\frac{S{S}_Y}{n}}$ denotes the standard deviation of Y scores, n denotes the number of pairs measured, $S{S}_Y$ denotes the sums of squares for scores Y, $M_{Y_1}$ denotes the mean of Y scores at first level, $M_{Y_2}$ denotes the mean of Y scores at second level.

Justification: The factors in the study are ‘Veteran (yes, no)’ and ‘level of patriotism indicated on a rating scale’. The factor ‘Veteran’ is dichotomous which has two levels ‘yes, no’ and factor ‘level of patriotism indicated on a rating scale’ is measured on nominal scale. The Point-biserial correlation coefficient is used to measure the relationship when one factor is dichotomous and other is measured on nominal scale.

Hence, Point-biserial correlation coefficient that should be used when studying relation between Veteran (yes, no) and level of patriotism indicated on a rating scale.

4.

To determine

Identify the correlation coefficient that should be used when studying relation between hierarchical ranking of a litter of mice for play and social behaviour.

Answer to Problem 12CAP

Spearman correlation coefficient should be used when studying relation between hierarchical ranking of a litter of mice for play and social behaviour.

Explanation of Solution

The given information is that, correlation coefficient between the hierarchical ranking of a litter of mice for play and social behaviour is to be studied.

Spearman correlation coefficient:

When the two factors are measured on either ranked scale or ordinal scale, then the strength and direction of the linear relationship between the two factors can be measured by Spearman correlation coefficient. This is also termed as Spearman rank-order correlation coefficient. The formula is,

$r_s=1-\frac{6{\displaystyle \sum D^2}}{n\left(n^2-1\right)}$

In the formula, D denotes the difference of ranks for factor X and factor Y, n denotes the sample size.

Justification: The factors in the study are ‘ranking of a litter of mice for play’ and ‘ranking of a litter of mice for social behaviour’. Both the factors are ranked on a ranking scale. The Spearman correlation coefficient is used to measure the relationship when two factors are ranked.

Hence, Spearman correlation coefficient should be used when studying relation between hierarchical ranking of a litter of mice for play and social behaviour.