Chapter 25, Problem 20E

(a)

To determine

To explain do these results provide evidence that the program is worthwhile.

Answer to Problem 20E

There is no sufficient evidence to support the claim that the program is worthwhile.

Explanation of Solution

The table for the Math scores is given for the students who repeated the exam again in August. It is also given that:

  $n=6$

And we will find the difference between the two samples as:

  

Now, the sample mean and the standard deviation of the difference are as follows:

  $\begin{array}{l}\overline{d}=\frac{4-16-6+2-6-10}{6}\\ =-5.3333\\ s_d=\sqrt{\frac{{(4-(-5.3333))}^2+{(-16-(-5.3333))}^2+{(-6-(-5.3333))}^2+\mathrm{....}+{(-10-(-5.3333))}^2}{6-1}}\\ =7.4476\end{array}$

Let us now define the hypotheses as:

  $\begin{array}{l}{H}_0:{\mu }_d=0\\ H_a:{\mu }_d<0\end{array}$

Now the degree of freedom is then as:

  $\begin{array}{l}df=n-1\\ =6-1\\ =5\end{array}$

Thus, the value of the test statistics is then as:

  $\begin{array}{l}t=\frac{\overline{d}}{s_d/\sqrt{n}}\\ =\frac{-5.3333}{7.4476/\sqrt{6}}\\ =-1.75\end{array}$

Thus, the P-value can be calculated using table T of appendix F as:

  $0.05<P<0.10$

As we know that the P-value is less than the significance level then the null hypothesis is rejected. Thus, we have,

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

Thus, we conclude that there is no sufficient evidence to support the claim that the program is worthwhile.

(b)

To determine

To find out which type of error was made if the conclusion is incorrect.

Answer to Problem 20E

If our conclusion turns out to be incorrect then it will lead to the Type II error.

Explanation of Solution

The table for the Math scores is given for the students who repeated the exam again in August. And from part (a) it is given that,

As we know that the P-value is less than the significance level then the null hypothesis is rejected. Thus, we have,

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

Thus, we conclude that there is no sufficient evidence to support the claim that the program is worthwhile. Thus,

Type I error: Reject the null hypothesis when the null hypothesis is true.

Type II error: Fail to reject the null hypothesis when the null hypothesis is false.

Thus, if our conclusion turns out to be incorrect then it will lead to the Type II error as we are failing to reject the null hypothesis.