Question

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then the conclusion about the null hypothesis, as well as the final conclusion that address the original claim. Assume that a simple random sample is selected from a normally distributed population.

Birth Weights A simple random sample of birth weights of 30 girls has a standard deviation of 829.5 hg. Use a 0.01 significance level to test the claim that birth weights of girls have the same standard deviation as birth weights of boys, which is 660.2 hg (based on Data Set 4 “Births” in Appendix B).

Short answer

The hypotheses are as follows.

$$\begin{gathered} H_0:\sigma =660.2 \\ {H}_1:\sigma \ne 660.2 \end{gathered}$$

The test statistic ${\chi }^2=45.78$, and the critical values are 52.336 and 13.121.

The null hypothesis is failed to be rejected.

There is sufficient evidence to support the claim that the standard deviation of the birth weights of girls is equal to that of the birth weights of boys.

Step-by-step solution

Given information 

The standard deviation of the birth weights for 30 girls is 829.5 hg.

The level of significance is 0.01 to test the claim that both girls and boys have the same standard deviation of birth weights, which is 660.2 hg.

State the hypotheses

To test the claim that the birth weights of girls have the same standard deviation as the birth weights of boys, the null and alternative hypotheses are formulated as follows.

$$\begin{gathered} H_0:\sigma =660.2 \\ {H}_1:\sigma \ne 660.2 \end{gathered}$$

Here, $\sigma$is the true standard deviation of the birth weights for girls.

State the test statistic 

The test statistic ${\chi }^2$with degrees of freedom $\left(n-1\right)$is given as follows.

.$$\begin{gathered} {\chi }^2=\frac{\left(n-1\right)s^2}{{\sigma }^2} \\ =\frac{\left(30-1\right){\left(829.5\right)}^2}{{\left(660.2\right)}^2} \\ =45.7804 \end{gathered}$$

The degree of freedom is computed as follows.

$$\begin{gathered} df=n-1 \\ =30-1 \\ =29 \end{gathered}$$

Thus, the test statistic is 45.78 with 29 degrees of freedom.

State the critical values 

The test is two-tailed.

The critical values are ${{\chi }^2}_L,{{\chi }^2}_R$, such that

$$\begin{gathered} P\left({\chi }^2<{{\chi }^2}_L\right)=\frac{0.01}{2}\left(0.005\right) \\ P\left({\chi }^2>{{\chi }^2}_L\right)=0.995 \\ P\left({\chi }^2>{{\chi }^2}_R\right)=\frac{0.01}{2}\left(0.005\right) \end{gathered}$$

Using the chi-square table for 29 degrees of freedom and a 0.005 level of significance, the right-tailed critical value is 52.336, and the left-tailed one is 13.121.

Thus,

$$\begin{gathered} {\chi }_L^2=13.121 \\ {\chi }_R^2=52.336 \end{gathered}$$

State the decision 

The decision rule states the following:

If the test statistic lies between the critical values, the null hypothesis will fail to be rejected; otherwise, it will be rejected.

In this case, the test statistic lies between the critical values, and hence, the null hypothesis is failed to be rejected at a 0.01 level of significance.

Thus, it can be concluded that there is sufficient evidence to support the claim that the standard deviation of the birth weights of girls is equal to the standard deviation of the birth weights of boys.