Chapter 23, Problem 38E

To determine

To find out should the company adopt the new tread pattern and test an appropriate test statistics also state your conclusion and also explain how you dealt with the outlier and why you made the recommendations you did.

Answer to Problem 38E

In the case of with outlier, there is the sufficient evidence that the company should adopt the new tread pattern.

And in the case of without outlier, there is the sufficient evidence that the company should not adopt the new tread pattern.

Explanation of Solution

In the question, it is given that a tier company is considering a newly designed tread pattern for its all-weather tires. Now, we will consider two cases: With outlier and without outlier. First let us consider the case of with outlier. Thus, we have,

  $n=10$

Now, the mean of the data on a test track will be as:

  $\begin{array}{l}\overline{x}=\frac{129+128+130+132+135+123+102+125+128+130}{10}\\ =126.2\end{array}$

And the standard deviation of the data on a test track will be as:

  $\begin{array}{l}s=\sqrt{\frac{{(129-126.2)}^2+\mathrm{....}+{(130-126.2)}^2}{10-1}}\\ =9.1384\end{array}$

Now, let us define the hypotheses:

  $\begin{array}{l}{H}_0:\mu =125\\ H_1:\mu \ne 125\end{array}$

Thus the value of the test statistics will be:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{126.2-125}{\frac{9.1384}{\sqrt{10}}}\\ =0.42\end{array}$

The degrees of freedom will be as according to the sample given:

  $\begin{array}{l}df=n-1\\ =10-1\\ =9\end{array}$

Now the P-value according to the degrees of freedom will be as:

  $P>0.20$

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

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

Thus, we conclude in this case is that there is the sufficient evidence that the company should adopt the new tread pattern.

Now, we will consider the second case of without outlier. Thus, we have,

  $n=9$

Now, the mean of the data on a test track will be as:

  $\begin{array}{l}\overline{x}=\frac{129+128+130+132+135+123+\mathrm{....}+130}{9}\\ =128.8889\end{array}$

And the standard deviation of the data on a test track will be as:

  $\begin{array}{l}s=\sqrt{\frac{{(129-128.8889)}^2+\mathrm{....}+{(130-128.8889)}^2}{9-1}}\\ =3.5512\end{array}$

Now, let us define the hypotheses:

  $\begin{array}{l}{H}_0:\mu =125\\ H_1:\mu \ne 125\end{array}$

Thus the value of the test statistics will be:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{128.8889-125}{\frac{3.5512}{\sqrt{9}}}\\ =3.29\end{array}$

The degrees of freedom will be as according to the sample given:

  $\begin{array}{l}df=n-1\\ =9-1\\ =8\end{array}$

Now the P-value according to the degrees of freedom will be as:

  $P>0.01$

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

  $P<0.05=5\%\Rightarrow \text{Reject }{H}_0$

Thus, we conclude in this case is that there is the sufficient evidence that the company should not adopt the new tread pattern.