Chapter 24, Problem 29E

(a)

To determine

To explain have the science scores declined significantly and cite appropriate statistical evidence to support your conclusion.

Answer to Problem 29E

Yes, the science score have declined significantly.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=147\\ {\overline{x}}_2=150\\ n_1=15109\\ n_2=7537\\ s_1=1.22\\ s_2=1.22\end{array}$

Now, let us define the hypotheses for testing:

  $\begin{array}{l}{H}_0:{\mu }_1={\mu }_2\\ H_1:{\mu }_1<{\mu }_2\end{array}$

Thus, the test statistics value is:

  $\begin{array}{l}t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{147-150}{\sqrt{\frac{{1.22}^2}{15109}+\frac{{1.22}^2}{7537}}}\\ =-174.37\end{array}$

And the degree of freedom is:

  $\begin{array}{l}df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})}{\frac{{(\raisebox{1ex}{$s_1^2$}\!\left/ \!\raisebox{-1ex}{$n_1$}\right.)}^2}{n_1-1}+\frac{{(\raisebox{1ex}{$s_2^2$}\!\left/ \!\raisebox{-1ex}{$n_2$}\right.)}^2}{n_2-1}}\\ =15060>1000\end{array}$

Thus, the P-value will be as:

  $P<0.005$

And as we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected. So,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is sufficient evidence to support the claim of decline.

(b)

To determine

To explain does this makes the results more convincing or less.

Answer to Problem 29E

Neither of the two.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=147\\ {\overline{x}}_2=150\\ n_1=15109\\ n_2=7537\\ s_1=1.22\\ s_2=1.22\end{array}$

And we conclude that there is sufficient evidence to support the claim of decline. Thus, this makes the results neither more convincing nor less convincing because both samples are very large and thus the actual sample size does not matter.