Chapter 23, Problem 34E

(a)

To determine

To find out what kind of error would he have made.

Answer to Problem 34E

It is Type II error.

Explanation of Solution

In the question, Bjork Larsen was trying to decide whether to use a new racing wax for cross country skis. He decided that the wax would be worth the price if he could average less than $55$ seconds on a course he knew well. Now, we define the hypotheses that:

  $\begin{array}{l}{H}_0:\mu =55\\ H_1:\mu <55\end{array}$

And the Type I error is when we reject the null hypothesis when it is true. And Type II error is when we fail to reject the null hypothesis when it is false. Thus, if he concludes not to buy the wax while the goal of an average of less than $55$ seconds was obtained then we failed to reject the null hypothesis while it was false and thus we made a Type II error.

(b)

To determine

To explain should he buy the wax or not.

Answer to Problem 34E

There is not sufficient evidence to buy wax.

Explanation of Solution

In the question, Bjork Larsen was trying to decide whether to use a new racing wax for cross country skis. He decided that the wax would be worth the price if he could average less than $55$ seconds on a course he knew well. Now, we define the hypotheses that:

  $\begin{array}{l}{H}_0:\mu =55\\ H_1:\mu <55\end{array}$

  $n=8$

The data is as:

  $\{56.3,65.9,50.5,52.4,46.5,57.8\}$

The mean of the data is as:

  $\begin{array}{l}\overline{x}=\frac{56.3+65.9+50.5+52.4+46.5+57.8}{8}\\ =53.1\end{array}$

The standard deviation of the data is as:

  $\begin{array}{l}s=\sqrt{\frac{{(56.3-53.1)}^2+\mathrm{....}+{(57.8-53.1)}^2}{8-1}}\\ =7.0285\end{array}$

The test statistic is as:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{53.1-55}{\frac{7.0285}{\sqrt{8}}}\\ =-0.76\end{array}$

The degrees of freedom is:

  $\begin{array}{l}df=n-1\\ =8-1\\ =7\end{array}$

Thus, the P-value will be:

  $P>0.10$

If the P-value is smaller than the significance level then the null hypothesis is rejected. Thus, we have,

  $P>0.05=5\%\Rightarrow \text{Fail to Reject }{H}_0$

Thus, we conclude that there is not sufficient evidence to buy wax.