Chapter 10.2, Problem 66E

(a)

To determine

To carry out an appropriate significance test and find out what conclusion would you draw.

Answer to Problem 66E

There is no convincing evidence that the new pressure assisted toilet reduces the average amount of water used by more than $0.5$ gallons per flush compare to its current model.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=1.64\\ {\overline{x}}_2=1.09\\ n_1=30\\ n_2=30\\ s_1=0.29\\ s_2=0.18\end{array}$

  $\alpha =0.05$

Therefore, the appropriate hypotheses for this is:

  $\begin{array}{l}{H}_0:{\mu }_1-{\mu }_2=0.5\\ H_a:{\mu }_1-{\mu }_2>0.5\end{array}$

Now, find the test statistics:

  $\begin{array}{l}t=\frac{({\overline{x}}_1-{\overline{x}}_2)-({\mu }_1-{\mu }_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{1.64-1.09-(0.5)}{\sqrt{\frac{{0.29}^2}{30}+\frac{{0.18}^2}{30}}}\\ =0.802\end{array}$

Now, the degree of freedom will be:

  $\begin{array}{l}df=\min (n_1-1,n_2-1)\\ =\min (30-1,30-1)\\ =29\end{array}$

So the P -value will be:

  $0.20<P<0.25$

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

  $P>0.05\Rightarrow \text{Fail to Reject }{H}_0$

Thus, we conclude that there is no convincing evidence that the new pressure assisted toilet reduces the average amount of water used by more than $0.5$ gallons per flush compare to its current model.

(d)

To determine

To explain what type of error could you have made- Type I error or Type II error, based on the conclusion in part (b).

Answer to Problem 66E

Type II error.

Explanation of Solution

We conclude in part (b) that,

There is no convincing evidence that the new pressure assisted toilet reduces the average amount of water used by more than $0.5$ gallons per flush compare to its current model.

A type I error occurs if we reject a null hypothesis when the null hypothesis is true. And the Type II error occurs if we fails to reject the null hypothesis when the null hypothesis is false.

Thus, in this case we fail to reject the null hypothesis then it is a Type II error.