Chapter 24, Problem 33E

To determine

To explain is there evidence that balls hit off the Stinger tees would have a higher initial velocity.

Answer to Problem 33E

There is evidence that balls hit off the Stinger tees would have a higher initial velocity.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=127\\ {\overline{x}}_2=128.83\\ n_1=6\\ n_2=6\\ s_1=0.89\\ s_2=0.41\end{array}$

The mean and the standard deviation is calculated by using excel by going to the data tab and then to the analysis tab and then selecting the data analysis option. Then we go to the t -test option in the dialogue box that appeared on screen. Then we select the data and the result will come out.

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 value of test statistics is:

  $\begin{array}{l}t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{127-128.83}{\sqrt{\frac{{0.89}^2}{6}+\frac{{0.41}^2}{6}}}\\ =-4.57\end{array}$

And the value of degree of freedom is as:

  $\begin{array}{l}df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})}{\frac{{(\raisebox{1ex}{$s_1^2$}\!\left/ \!\raisebox{-1ex}{$n_1$}\right.)}^2}{n_1-1}+\frac{{(\raisebox{1ex}{$s_2^2$}\!\left/ \!\raisebox{-1ex}{$n_2$}\right.)}^2}{n_2-1}}\\ =7\end{array}$

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 a higher initial velocity.