Chapter 25, Problem 27E

(a)

To determine

To explain is there evidence that after training players can throw strikes than $60\%$ of the time.

Answer to Problem 27E

There is sufficient evidence to support the claim that after training players can throw strikes than $60\%$ of the time.

Explanation of Solution

The table of number of strikes before and after is given in the question. It is also given that,

  $n=20$

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

  $\begin{array}{l}\overline{x}=\frac{35+36+\mathrm{....}+35+32}{20}\\ =33.15\\ s=\sqrt{\frac{{(35-33.15)}^2+{(36-33.15)}^2+\mathrm{....}+{(35-33.15)}^2+{(32-33.15)}^2}{20-1}}\\ =2.3232\end{array}$

Let us now define the hypotheses as:

  $\begin{array}{l}{H}_0:\mu =60\%\times 50=30\\ H_a:\mu >30\end{array}$

Now the degree of freedom is then as:

  $\begin{array}{l}df=n-1\\ =20-1\\ =19\end{array}$

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

  $\begin{array}{l}t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}}\\ =\frac{33.15-30}{\frac{2.3232}{\sqrt{20}}}\\ =6.064\end{array}$

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

  $P<0.005$

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

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is sufficient evidence to support the claim that after training players can throw strikes than $60\%$ of the time.

(b)

To determine

To explain is there evidence that the training is effective in improving a player’s ability to throw strikes.

Answer to Problem 27E

There is no sufficient evidence to support the claim that the training is effective in improving a player’s ability to throw strikes.

Explanation of Solution

The table of number of strikes before and after is given in the question. It is also given that,

  $n=20$

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:

  $\begin{array}{l}\overline{d}=\frac{-7-7+\mathrm{....}+2+5}{20}\\ =-0.1\\ s_d=\sqrt{\frac{{(-7-(-0.1))}^2+{(-7-(-0.1))}^2+\mathrm{....}+{(2-(-0.1))}^2+{(5-(-0.1))}^2}{20-1}}\\ =3.3230\end{array}$

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:

  $\begin{array}{l}df=n-1\\ =20-1\\ =19\end{array}$

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

  $\begin{array}{l}t=\frac{\overline{d}}{s_d/\sqrt{n}}\\ =\frac{-0.1}{3.3230/\sqrt{20}}\\ =-0.135\end{array}$

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

  $P>0.20$

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 training is effective in improving a player’s ability to throw strikes.