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# Modern medical practice tells us not to encourage babies to become too fat. Is there a positive correlation between the weight x of a 1-year-old baby and the weighty of the mature adult (30 years old)? A random sample of medical files produced the following information for 14 females:$\scriptstyle\begin{matrix} \text{x (lb)} & \text{21} & \text{25} & \text{23} & \text{24} & \text{20} & \text{15} & \text{25} & \text{21} & \text{17} & \text{24} & \text{26} & \text{22} & \text{18} & \text{19}\\ \text{y (lb)} & \text{125} & \text{125} & \text{120} & \text{125} & \text{130} & \text{120} & \text{145} & \text{130} & \text{130} & \text{130} & \text{130} & \text{140} & \text{110} & \text{115}\\ \end{matrix}$$\Sigma x=300$; $\Sigma y=1775$; $\Sigma x^{2}=6572$; $\Sigma y^{2}=226,125$; $\Sigma x y=38,220$ Test the claim that the population correlation coefficient $\boldsymbol{\rho}$ is positive at the 1% level of significance.

Solution

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Step 1
1 of 2

Given:

$\alpha=1\%=0.01$

$n=14$

Result part c:

$r=0.4680$

Determine the hypotheses:

$H_0:\rho=0$

$H_1\rho>0$

The value of the test statistic is:

$t=\dfrac{r}{\sqrt{(1-r^2)/(n-2)}}=\dfrac{0.4680}{\sqrt{(1-0.4680^2)/(14-2)}}\approx 1.835$

Determine the corresponding probability (P-value) using table 6 with $df=n-2=14-2=12$ (one-tail area):

$0.025

If the P-value is smaller than the significance level, reject the null hypothesis:

$P>0.01\Rightarrow \text{ Reject } H_0$

There is not sufficient evidence to support the claim that the population correlation coefficient is positive.

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