Chapter 9.1, Problem 28E

(a)

To determine

To Explain: the appropriate hypotheses for performing significance test.

Answer to Problem 28E

  $H_0:\mu =6.9$

  $H_1:\mu >6.9$

Explanation of Solution

Given:

Claim is that mean is more than 6.9

The null hypothesis statement is that the population value is equal to the value given in the claim:

  $H_0:\mu =6.9$

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis statement is that the population proportion 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.

  $H_1:\mu >6.9$

  $\mu$ is the mean number of new words

(b)

To determine

To Explain: the P-value using the simulation results.

Answer to Problem 28E

0.06

There is a 6% possibility that the mean number of new words in a sample of size 5 is 9.2 or larger, when the true mean number of new words in the population is actually 6.9.

Explanation of Solution

Given:

  $\overline{x}=9.2$

Claim is that mean is greater than 6.9.

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

  $\begin{array}{l}{H}_0:\mu =6.9\\ H_1:\mu >6.9\end{array}$

The P-value is the probability of getting the value of the test statistic, or a value more extreme, when the null hypothesis is true.

12 of the 200 dots in the dot plot are at 9.2 or to its right.

  $\text{P-value =}\frac{12}{200}=\frac{3}{50}=0.06=6\%$

There is a 6% possibility that the mean number of new words in a sample of size 5 is 9.2 or larger, when the true mean number of new words in the population is actually 6.9.

(c)

To determine

To Explain: the conclusion would make.

Answer to Problem 28E

There is no sufficient convincing evidence that the mean number of words is greater than 6.9.

Explanation of Solution

Given:

  $\overline{x}=9.2$

Given claim: mean is greater than 6.9.

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

  $\begin{array}{l}{H}_0:\mu =6.9\\ H_1:\mu >6.9\end{array}$

The P-value is the probability of getting the value of the test statistic, or a value more extreme, when the null hypothesis is true.

12 of the 200 dots in the dot plot are at 9.2 or to its right.

  $\text{P-value =}\frac{12}{200}=\frac{3}{50}=0.06$

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

Let us use the significance level $\alpha =0.05$ (which is the most commonly used significance level).

  $0.06>0.05\Rightarrow \text{Fail to reject }{H}_0$

There is no sufficient convincing evidence that the mean number of words is greater than 6.9.