Chapter 10.2, Problem 65E

(a)

To determine

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

Answer to Problem 65E

There is no convincing evidence that the true mean cholesterol reduction with the new drug is more than ten milligrams per deciliter of blood greater than the mean cholesterol reduction with the current drug.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=54.1\\ {\overline{x}}_2=68.7\\ n_1=14\\ n_2=15\\ s_1=11.93\\ s_2=13.3\end{array}$

  $\alpha =0.05$

Therefore, the appropriate hypotheses for this is:

  $\begin{array}{l}{H}_0:{\mu }_1-{\mu }_2=-10\\ H_a:{\mu }_1-{\mu }_2<-10\end{array}$

Now, find the test statistics:

  $\begin{array}{l}t=\frac{({\overline{x}}_1-{\overline{x}}_2)-({\mu }_1-{\mu }_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{54.1-68.7-(-10)}{\sqrt{\frac{{11.93}^2}{14}+\frac{{13.3}^2}{15}}}\\ =-0.982\end{array}$

Now, the degree of freedom will be:

  $\begin{array}{l}df=\min (n_1-1,n_2-1)\\ =\min (14-1,15-1)\\ =13\end{array}$

So the P -value will be:

  $0.15<P<0.20$

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 true mean cholesterol reduction with the new drug is more than ten milligrams per deciliter of blood greater than the mean cholesterol reduction with the current drug.

(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 65E

Type II error.

Explanation of Solution

We conclude in part (b) that,

There is no convincing evidence that the true mean cholesterol reduction with the new drug is more than ten milligrams per deciliter of blood greater than the mean cholesterol reduction with the current drug.

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.