Chapter 12.1, Problem 19E

To determine

a.

To Calculate:

The correlation coefficient ‘r’.

Answer to Problem 19E

Solution:

The correlation coefficient is $r=0.375$.

Explanation of Solution

Given:

Mothers’ Birth Weights and Babies’ Birth Weights
Mother	lb	6	6	7	8	7	7	7	6	9	6	9	8	6	5	6	5
	oz	6	2	9	10	6	15	7	5	3	10	3	9	1	0	1	0
Baby	lb	5	6	7	7	10	5	9	5	8	7	9	8	5	7	7	8
	oz	11	14	10	2	0	13	0	9	13	11	10	6	8	15	2	5

Since one pound is equal to 16 ounce converting pounds (lb) to ounce (Oz) we have,

Mother	102	98	121		138	118	127		119		101	147
Baby	91	110	122		114	160	93		144		89	141
Mother	106	147		137		97		80		97		80
Baby	123	150		134		88		127		114		133

Formula:

The Pearson correlation coefficient for paired data from a sample is given by:

$r=\frac{n{\displaystyle \sum x_i{y}_i-({\displaystyle \sum x_i})({\displaystyle \sum y_i})}}{\sqrt{n{\displaystyle \sum x_i^2-{({\displaystyle \sum x_i})}^2}}\sqrt{n{\displaystyle \sum y_i^2-{({\displaystyle \sum y_i})}^2}}}$…(1)

Where n is the number of data pairs of the sample,

$x_i$ is the i^th value of the explanatory variable, and

$y_i$ is the i^th value of the response variable

Calculation:

Using the above table we have,

Mother weight (lb)xi	Babies weight (lb)yi	(xi)(yi)	xi 2	yi 2
102	91	9282	10404	8281
98	110	10780	9604	12100
121	122	14762	14641	14884
138	114	15732	19044	12996
118	160	18880	13924	25600
127	93	11811	16129	8649
119	144	17136	14161	20736
101	89	8989	10201	7921
147	141	20727	21609	19881
106	123	13038	11236	15129
147	154	22638	21609	23716
137	134	18358	18769	17956
97	88	8536	9409	7744
80	127	10160	6400	16129
97	114	11058	9409	12996
80	133	10640	6400	17689
${\displaystyle \sum x_i}$=1815	${\displaystyle \sum y_i}$=1937	${\displaystyle \sum x_i}{y}_i$=222527	${\displaystyle \sum x_i^2}$=212949	${\displaystyle \sum y_i^2}$=242407

Here n=16, Substituting the total values in (1),

$r=\frac{16(222527)-(1815)(1937)}{\sqrt{16(212949)-{(1815)}^2}\sqrt{16(242407)-{(1937)}^2}}$

$r=\frac{3560432-3515655}{\sqrt{3407184-3294225}\sqrt{3878512-3751969}}$

$r=\frac{44777}{\sqrt{112959}\sqrt{126543}}$

$r=\frac{44777}{336.0937\times 355.7255}$

$r=\frac{44777}{119558.2}$

$r=0.3745205264\approx 0.375$

Thus, the correlation coefficient is, $r=0.375$.

To determine

Whether the correlation coefficient is statistically significant at the 0.05 level of significance.

Answer to Problem 19E

Solution:

The correlation coefficient is not statistically significant at the level of significance $\alpha =0.05$ for the sample size $n=16$.

Explanation of Solution

Procedure:

The correlation coefficient $r$ is statistically significant, if the absolute value of the correlation coefficient is greater than or equal to the critical value, $r_{\alpha }$, from the table critical values of the Pearson correlation coefficient.

Where $\alpha$ level of significance and $n$ is the sample size.

On other hand,

A sample correlation coefficient, $r$, is statistically significant if $\left|r\right|\ge r_{\alpha }$.

Given:

$r=0.375,\alpha =0.05,n=16$

To find the critical value $r_{\alpha }$:

The critical value $r_{\alpha }$ for $\alpha =0.05$ with $n=16$ from the Pearson correlation coefficient table is $r_{\alpha }=0.497$.

Comparing this critical value to the absolute value of the correlation coefficient. We get the relation,

$0.375<0.497$

Thus, $\left|r\right|<r_{\alpha }$.

Therefore the correlation coefficient is not statistically significant at the level of significance $\alpha =0.05$ for the sample size $n=16$.

To determine

(c)

To Find:

The coefficient of determination, $r^2$.

Answer to Problem 19E

Solution:

The coefficient of determination $r^2=0.140$.

Explanation of Solution

Definition:

The coefficient of determination, $r^2$, is a measure of the proportion of the variation in the response variable $(y)$ that can be associated with the variation in the explanatory variable $(x)$.

Calculation:

Correlation Coefficient is $r=0.375$.

Coefficient of determination is the square of the Correlation Coefficient.

Then,

$\begin{array}{rll}{r}^2& =& {\left(0.375\right)}^2\\ & =& 0.140625\approx 0.140\end{array}$

Thus, the Coefficient of determination $r^2=0.140$.

To determine

(d)

To Interpret:

The Coefficient of determination $r^2$ for the given set of data.

Answer to Problem 19E

Solution:

The variation in the babies’ birth weights can be associated with the variation in birth weights of the mothers by $14.0\%$.

Explanation of Solution

The correlation coefficient for the babies’ birth weights with the birth weights of the mothers is $r=0.375$.

The coefficient of determination, $r^2$, is a measure of the proportion of the variation in the response variable (babies’ birth weights) that can be associated with the variation in the explanatory variable (birth weights of mothers).

Coefficient of determination is, $r^2=0.140$.

Interpretation:

Thus, approximately $14.0\%$ of the variation in the babies’ birth weights that can be associated with the variation in birth weights of the mothers.