Estriol Level and Birth Weight.The data from Exercise 14.42 for estriol levels of pregnant women and birth weights of their children are on the WeissStats site. Do the data provide sufficient evidence to conclude that estriol level and birth weight are positively linearly correlated?

The data from Exercise 14.42 for estriol levels of pregnant women and birth weights of their children are on the WeissStats site

Check whether or not it is reasonably apply the correlation $t$-test procedure by using the data from Exercise$14.42$

- From the residual plot versus estriol level, there is an increasing level of variability.

- From the normal probability plot of residuals, it is clear that the residuals are roughly linear pattern.

Hence, the assumption 2 for the regression inferences is violated for the variables weight and Estriol level. But the normality assumption for the regression inferences is not violated.

Therefore, it is reasonably applying the correlation $t$-test procedure for given data.

Check whether the data provide sufficient evidence to conclude that estriol level and birth weight are positively linearly correlated or not.

The test hypotheses are as follows:

Null hypothesis:

$H_0:\rho =0$

That is, the estriol level and birth weight are not positively linearly correlated.

Alternative hypothesis:

$H_a:\rho >0$

That is, the estriol level and birth weight are positively linearly correlated.

Obtain the correlation and $p$-value between estriol level and birth weight by using MINITAB.

MINITAB procedure:

Step 1: Select Stat >Basic Statistics > Correlation.

Step 2: In Variables, select estriol level and birth weight from the box on the left.

Step 3: ClickOK.

MINITAB output:

Use the significance level, $\alpha =0.05$.

Here, $p$-value is lesser than the level of significance.

That is, $p$-value $(=0)<\alpha (=0.05)$.

Therefore, by the rejection rule, it can be concluded that there is evidence to reject the null hypothesis $\left(H_0\right)$at $\alpha =0.05$.

Thus, the data provide sufficient evidence to conclude that estriol level and birth weight are positively linearly correlated at $5\%$significance level.