Chapter 24, Problem 32E

To determine

To explain do these results suggest that the mean times of seeded heats are not equal.

Answer to Problem 32E

Yes, these results suggest that the mean times of seeded heats are not equal.

Explanation of Solution

Let us the check the conditions and assumptions for inference for this case as:

Random condition: It is not satisfied because the heats were not determined at random.

Independent condition: It is satisfied because since no individual can be in more than on heat.

Normal condition: It is satisfied because the distributions are symmetric and have roughly the same variability.

Thus, all the conditions are not met.

Now, it is given that:

  $\begin{array}{l}{\overline{x}}_1=260.2263\\ {\overline{x}}_2=250.7838\\ n_1=8\\ n_2=8\\ s_1=3.0311\\ s_2=2.1485\end{array}$

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:

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

Thus, the value of test statistics is:

  $\begin{array}{l}t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{260.2263-250.7838}{\sqrt{\frac{{3.0311}^2}{8}+\frac{{2.1485}^2}{8}}}\\ =7.19\end{array}$

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$}\right.)}^2}{n_2-1}}\\ =12\end{array}$

Thus, the P-value will be as:

  $P<0.01$

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 difference.