Chapter 10, Problem 4CRE

To determine

To find: the appropriate statistical proof to justify the answer.

Answer to Problem 4CRE

No

Explanation of Solution

Given:

  $\begin{array}{c}{\overline{x}}_1=272.40\\ {\overline{x}}_2=274.73\\ s_1=59.2\\ s_2=57.5\\ n_1=840\\ n_2=1077\end{array}$

Formula used:

  $t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_1}}}$

Calculation:

The hypothesis

  $\begin{array}{c}{H}_0:{\mu }_1={\mu }_2\\ H_1:{\mu }_1\ne {\mu }_2\end{array}$

The test statistic:

  $\begin{array}{c}t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_1}}}\\ =\frac{272.40-274.72}{\sqrt{\frac{{59.2}^2}{840}}+\frac{{57.5}^2}{1077}}\\ =-0.866\end{array}$

The degrees of freedom:

  $\begin{array}{c}df=\text{min(}{n}_1-1,n_2-1)\\ =\text{min}\left(840-1,1077-1\right)\\ =839>100\end{array}$

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme.

  $\begin{array}{l}0.30=2\times 0.15<P<2\times 0.20=0.40\\ 0.30<P<0.40\end{array}$

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:

  $P>0.01=1\%\Rightarrow \text{Fail to reject }{H}_0$

There is not enough proof to help the claim that there is a difference.