Chapter 8.2, Problem 1AC

Car Thefts

You recently received a job with a company that manufactures an automobile antitheft device. To conduct an advertising campaign for the product, you need to make a claim about the number of automobile thefts per year. Since the population of various cities in the United States varies, you decide to use rates per 10,000 people. (The rates are based on the number of people living in the cities.) Your boss said that last year the theft rate per 10,000 people was 44 vehicles. You want to see if it has changed. The following are rates per 10,000 people for 36 randomly selected locations in the United States. Assume σ = 30.3.

Source: Based on information from the National Insurance Crime Bureau.

Using this information, answer these questions.

1. What hypotheses would you use?

2. Is the sample considered small or large?

3. What assumption must be met before the hypothesis test can be conducted?

4. Which probability distribution would you use?

5. Would you select a one- or two-tailed test? Why?

6. What critical value(s) would you use?

7. Conduct a hypothesis test.

8. What is your decision?

9. What is your conclusion?

10. Write a brief statement summarizing your conclusion.

11. If you lived in a city whose population was about 50,000, how many automobile thefts per year would you expect to occur?

To determine

To find: The hypotheses.

Answer to Problem 1AC

Hypotheses:

$H_0:\mu =44$

$H_1:\mu \ne 44$

Explanation of Solution

Given info:

The data shows the rates per 10,000 people for 36 randomly selected locations in the United States

Calculation:

Null hypothesis:

Null hypothesis is a statement about population parameter, its value is equal to the claim value, which is denoted by $H_0$. The population parameter can be mean, standard deviation, proportion, variance, and so forth. The possible symbols used in the null hypothesis would be $\le ,\ge$, or =.

Alternative hypothesis:

It is complementary to the null hypothesis. That is, it differs from the null hypothesis. The possible symbols used in the alternative hypothesis would be <,>, or ≠. It is denoted by $H_1$.

Here, the claim is that “the theft rate per 10,000 people was 44 vehicles”. This can be written as $\mu =44$. The complement of the claim is $\mu \ne 44$. In the given experiment, the null hypothesis indicates the claim.

The test hypotheses are given below:

Null hypothesis: $H_0:\mu =44$

Alternative hypothesis: $H_1:\mu \ne 44$

To determine

To check: The sample is considered small or large.

Explanation of Solution

Since the sample is greater than 30, that is $n\left(=36\right)>30$, the sample can be considered large.

To determine

To find: The assumption must be met before the hypothesis test can be conducted.

Explanation of Solution

The assumption must be met before the hypothesis test can be conducted is “the variable is normally distributed”.

To determine

To identify: The probability distribution.

Answer to Problem 1AC

The z-test should be used.

Explanation of Solution

Assumption for z test:

• Simple random sample from the population.
• The population standard deviation is known.
• Either the population is normally distributed or $n\ge 30$

Here, the population standard deviation is known and $n\ge 30$. Thus, the z-test should be used.

To determine

To check: Whether the given test is one tailed test or two tailed test.

Answer to Problem 1AC

The test is two tailed test.

Explanation of Solution

Here, the researcher interested to test whether the car theft rate per 10,000 people has changed or not. This indicates that the alternative hypothesis contains the not equal symbol. Therefore, the test is two tailed test.

To determine

To find: The critical value.

Answer to Problem 1AC

The critical value is ±1.96.

Explanation of Solution

Calculation:

Answer will vary. One of the possible answers is given below:

Assume that the level of significance $\alpha =0.05$.

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

From the output, the critical value is ±1.96.

To determine

To conduct: The hypotheses test.

Answer to Problem 1AC

The test value is 2.37.

Explanation of Solution

Calculation:

Software Procedure:

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

• Choose Stat > Basic Statistics > 1-Sample Z.
• In Samples in Column, enter the column of Rates.
• In Standard deviation, enter 30.3.
• In Perform hypothesis test, enter the test mean as 44.
• Check Options; enter Confidence level as 95%.
• Choose not equal in alternative.
• Click OK.

Output using the MINITAB software is given below:

From the output, the test value is 2.37.

To determine

To make: The decision.

Answer to Problem 1AC

The decision is “reject the null hypothesis”.

Explanation of Solution

Calculation:

Software Procedure:

Step-by-step procedure to indicate the appropriate area and critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Enter 2.37 under show reference lines at X values.
• Click OK.

Output using the MINITAB software is given below:

From the output, it can be observed that the test statistic value falls in the critical region. Therefore, the null hypothesis is rejected.

To determine

To describe: The conclusion.

Answer to Problem 1AC

The conclusion is that, the car theft rate per 10,000 people has changed.

Explanation of Solution

Justification:

Here, the null hypothesis is rejected. Therefore, there is enough evidence to support the claim that the theft rate per 10,000 people was 44 vehicles at 5% level of significance.

To determine

To write: The statement.

Explanation of Solution

Answer will vary. One of the possible answers is given below:

From the about results, it can be observed that the car theft rate per 10,000 people has changed. This signifies that the car theft rate per 10,000 people has increased.

To determine

To find: The number of automobile thefts per year would expect to occur.

Answer to Problem 1AC

The number of automobile thefts per year would expect to occur is approximately 280 car theft in the city.

Explanation of Solution

Calculation:

From the output obtained in part (7), the sample mean is 55.97.

The number of automobile thefts per year would expect to occur is,

$\begin{array}{c}\left(55.97\right)\left(\frac{50,000}{10,000}\right)=55.97\times 5\\ =\text{279}\text{.85}\end{array}$

Thus, the number of automobile thefts per year would expect to occur is approximately 280 car theft in the city.