Chapter 15, Problem 1SR

A recent insurance industry report indicated that 40% of those persons involved in minor traffic accidents this year have been involved in at least one other traffic accident in the last 5 years. An advisory group decided to investigate this claim, believing it was too large. A sample of 200 traffic accidents this year showed 74 persons were also involved in another accident within the last 5 years. Use the .01 significance level.

1. (a) Can we use z as the test statistic? Tell why or why not.
2. (b) State the null hypothesis and the alternate hypothesis.
3. (c) Show the decision rule graphically.
4. (d) Compute the value of z and state your decision regarding the null hypothesis.
5. (e) Determine and interpret the p-value.

To determine

Check whether people can use z as the test statistic and explain the reason.

Answer to Problem 1SR

Yes, people can use z as the test statistic because both nπ and n (1–π) exceed 5.

Explanation of Solution

Requirements to check:

$\begin{array}{l}n\pi >5\\ n\left(1-\pi \right)>5\end{array}$

It is given that the sample size n is 100.

For $n\pi >5$:

$\begin{array}{c}n\pi =0.70\times 100\\ =70\\ >5\end{array}$

For $n\left(1-\pi \right)>5$:

$\begin{array}{c}n\pi =100\times \left(1-0.70\right)\\ =30\\ >5\end{array}$

Hence, the requirements are satisfied for using the z-statistic as the binomial distribution.

To determine

State the null and alternate hypotheses.

Explanation of Solution

In this case, the test is to check whether less than 40% of the persons involved in minor traffic accidents this year have been involved in at least one other traffic accident in the last 5 years.

Let π represents population proportion of persons involved in minor traffic accidents this year have been involved in at least one other traffic accident in the last 5 years.

Therefore, the null and alternate hypotheses are shown below:

$\begin{array}{l}{H}_0:\pi \ge 0.40\\ H_1:\pi <0.40\end{array}$

To determine

Show the decision rule graphically.

Explanation of Solution

Step-by-step procedure to show the decision rule graphically using MINITAB software:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0 and Standard deviation as 1.
• Click the Shaded Area tab.
• Choose Probability and Left Tail for the region of the curve to shade.
• Enter the Probability as 0.01.
• Click OK.

Output using MINITAB software is obtained as follows:

From the output, the critical value is –2.326.

Therefore, the decision rule is rejecting the null hypothesis if test statistic is less than –2.326.

To determine

Find the value of z-statistic and the write the decision regarding the null hypothesis.

Answer to Problem 1SR

The value of chi-square is –0.87.

Explanation of Solution

Calculation:

The sample size n is 200 and x is 74.

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

• Choose Stat > Basic Statistics > 1 Proportion.
• Choose Summarized data.
• In Number of events, enter 74. In Number of trials, enter 200.
• Enter Hypothesized proportion as 0.40.
• Check Options, enter Confidence level as 99.0.
• Choose less than in alternative.
• Select Method as Normal approximation.
• Click OK in all dialogue boxes.

Output is obtained as follows:

From the output, the value of the test statistic is –0.87.

In this case, the critical values is –2.326 and the test statistic is –0.87.

Here, the test statistic value is less than the critical value.

That is, –0.87 > –2.326.

Therefore, do not reject the null hypothesis.

To determine

Find and interpret the p-value.

Explanation of Solution

From the output of Part (d), it can be observed that the p-value is 0.193 and it is more than the level of significance. Therefore, there is no sufficient evidence to conclude that less than 40% of the persons involved in minor traffic accidents this year have been involved in at least one other traffic accident in the last 5 years.