Chapter 11, Problem 11.1.1RE

For Exercises 1 through 10, follow these steps.

a. State the hypotheses and identify the claim.

b. Find the critical value(s).

c. Compute the test value.

d. Make the decision.

e. Summarize the results.

Use the traditional method of hypothesis testing unless otherwise specified. Assume all assumptions have been met.

1. Traffic Accident Fatalities A traffic safety report indicated that for the 21–24 year age group, 31.58% of traffic fatalities were victims who had used a seat belt. Victims who were not wearing a seat belt accounted for 59.83% of the deaths, and the status of the rest was unknown. A study of 120 randomly selected traffic fatalities in a particular region showed that for this age group, 35 of the victims had used a seat belt, 78 had not, and the status of the rest was unknown. At α = 0.05, is there sufficient evidence that the proportions differ from those in the report?

Source: New York Times Almanac.

To determine

To state: The hypothesis and the claim.

Answer to Problem 11.1.1RE

The null and alternative hypotheses are:

$H_0$: The proportion is same as the reported proportion.

$H_1$:  The proportion is different from the reported proportion.

And the claim of the test is the observed proportion is different from the reported proportion.

Explanation of Solution

Given info:

The percentage of death and the observed count corresponding to each reason are provided in the question. The level of significance is $\alpha =0.05$.

Justification:

The researcher wants to know that whether the proportion of the traffic fatalities corresponding to each reason is different from the reported proportion or not. The null and alternative hypothesis can be defined as:

Null hypothesis:

$H_0$: The proportion is same as the reported proportion.

Alternative hypothesis:

$H_1$: The proportion is different from the reported proportion.

In the provided situation, the claim of the study will be the observed proportion corresponding to each reason is different from the reported proportion.

To determine

The critical value.

Answer to Problem 11.1.1RE

The required critical value is 5.991.

Explanation of Solution

The required critical value is obtained from the provided chi-square table in the textbook. The number of categories is 3.

The degrees of freedom is calculated as:

$\begin{array}{c}\text{Degrees of freedom}=\text{Number of categories}-1\\ =3-1\\ =2\end{array}$

Therefore, the critical value at $\alpha =0.05$ level of significance and 2 degrees of freedom is obtained as 5.991.

To determine

The value of the test statistic.

Answer to Problem 11.1.1RE

The test statistic value is 1.819.

Explanation of Solution

Calculation:

Software procedure:

Step-by-step procedure to obtain the test statistic using the MINITAB software:

• Enter the data in the Minitab worksheet.
• Go to Stat> Tables> Chi-Square Goodness-of-Fit Test (one variable).
• Specify the “Observed count”, choose the option “Proportions specified by historic count”, and specify the column where the percentage is written.
• Click on OK.

Output using the MINITAB software is given below:

Therefore, the obtained value of the test statistic is 1.819.

To determine

To make: The decision.

Answer to Problem 11.1.1RE

The null hypothesis will not be rejected.

Explanation of Solution

The obtained value of the test statistic is 1.819 and the critical value is 5.991. As the obtained value of the chi-square statistic is less than the critical value, it can be said that there is not enough evidence to reject the null hypothesis at $\alpha =0.05$.

To determine

To summarize: The results.

Answer to Problem 11.1.1RE

According to the obtained result, the claim of the study is not true.

Explanation of Solution

The null hypothesis is not rejected. On the basis of the obtained result, it can be concluded that the result of the proportion of the traffic fatalities is not different from the reported proportion at $\alpha =0.05$. That is, the claim is not true.