Chapter 9.3, Problem 73E

(a)

To determine

To Explain: that there is convincing evidence at the $\alpha =0.01$ significance level that the average speed of drivers in this construction zone is greater than the posted speed limit.

Answer to Problem 73E

There is enough convincing proof that the mean speed of drivers in this construction zone is bigger than the posted speed limit.

Explanation of Solution

Given:

  $\begin{array}{l}\alpha =0.01\\ n=10\end{array}$

Formula used:

  $t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$

Calculation:

Conditions

The three conditions are: Random, independent.(10% condition), Normal/Large sample.

Independent: satisfied, because the sample of 10 drivers is less than 10% of the population of all drivers.

Normal/ Large sample: satisfied, because the pattern in the normal quintile plot is roughly linear, this indicates that the distribution is approximately Normal

Since all conditions are satisfied, it is suitable to perform a hypothesis test for the population mean.

The mean is

  $\begin{array}{c}\overline{x}=\frac{\begin{array}{l}27+33+32+21+30\\ +30+29+25+27+34\end{array}}{10}\\ =\frac{288}{10}\\ =28.8\end{array}$

The variance is

  $\begin{array}{c}s=\sqrt{\frac{\begin{array}{l}{\left(27-28.8\right)}^2+{\left(33-28.8\right)}^2+{\left(32-28.8\right)}^2+{\left(21-28.8\right)}^2+{\left(30-28.8\right)}^2\\ +{\left(30-28.8\right)}^2+{\left(29-28.8\right)}^2+{\left(25-28.8\right)}^2+{\left(27-28.8\right)}^2+{\left(34-28.8\right)}^2\end{array}}{10-1}}\\ =3.9384\end{array}$

Hypothesis test

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis statement is the population mean is equal to the value given in the claim. if the null hypothesis is the claim, then the alternative hypothesis statement is the opposite of the null hypothesis.

  $\begin{array}{c}{H}_0:\mu =25\\ H_1:\mu >25\end{array}$

The statistic is

  $\begin{array}{c}t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}\\ =\frac{28.8-25}{3.9384/\sqrt{10}}\\ =3.051\end{array}$

The P-value is the probability of getting the value of the test static, or a value more extreme, assuming that the null hypothesis is true $df=n-1=10-1=9$ .

  $0.005<P<0.01$

Command Ti83/84- calculator: tcdf (3.051, 1E99, 9) which will return a P-value of 0.00689 Note: it could replace 1E99 by any other very large positive number.

If the P-value is lesser than the significance level $\alpha$ , then the null hypothesis is rejected.

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

There is enough convincing proof that the mean speed of drivers in this construction zone is bigger than the posted speed limit.

(b)

To determine

To Explain: the conclusion in part (a), which kind of mistake a Type-I error or a Type-II error could made, explain this mistake would mean in context.

Answer to Problem 73E

There is enough convincing proof that the average speed of drivers in this construction zone is bigger than the posted speed limit

Explanation of Solution

In part (a), rejected the null hypothesis $H_0$ .

Type I error: reject the null hypothesis $H_0$ , once the null hypothesis is true.

Type II error: Fail to reject the null hypothesis $H_0$ , once the null hypothesis is false

Since we reject the null hypothesis $H_0$ , it is only possible to have made a type I error.

This would mean that there is enough convincing proof that the average speed of drivers in this construction zone is bigger than the posted speed limit, whereas the average speed is actually the posted speed limit.