Assume that you recently recived 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 tefts per year. Since the population of various cities 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 teft 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.
A. What hypothesis would you use?
B. Is the sample considered small or large?
C. What assumption must be met before the hypothesis test can be conducted ?
D. Which probability distribution would you use?
E. Would you select a one-or two-tailed test? why?
F. Conduct a hypothesis test. Use Standard deviation =30.3
G. If you lived in a city whose population was about 50,000, how many automobile thefts per year would you expect to occur?
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what hypotheses would
you use?
A. The hypothesis would be that the theft rate per 10,000 people has not changed since last year.
B. The sample is considered small since it consists of 36 randomly selected locations.
C. The assumption that must be met before the hypothesis test can be conducted is that the theft rates follow a normal distribution.
D. The probability distribution that would be used is the normal distribution.
E. In this case, a one-tailed test should be selected because the hypothesis is focused on whether the theft rate has decreased or stayed the same. There is no interest in testing whether it has increased.
F. To conduct a hypothesis test, we can use the following steps:
1. Step 1: Set up the hypotheses:
- Null hypothesis (H0): The theft rate per 10,000 people has not changed since last year (µ = 44).
- Alternative hypothesis (Ha): The theft rate per 10,000 people has changed since last year (µ ≠ 44).
2. Step 2: Select the significance level (α). Let's assume it is 0.05 (5%).
3. Step 3: Calculate the test statistic:
- In this case, we can use a Z-test statistic.
- The formula for the test statistic is: Z = (x̄ - µ) / (σ / √n)
- x̄: Sample mean of the theft rates
- µ: Hypothesized population mean (44)
- σ: Standard deviation (30.3)
- n: Sample size (36)
4. Step 4: Determine the critical value(s) or the p-value:
- We will compare the calculated test statistic (Z) to the critical value(s) from the standard normal distribution table or calculate the p-value.
5. Step 5: Make a decision and interpret the result:
- If the calculated test statistic is greater than the critical value(s) or the p-value is less than the significance level, we reject the null hypothesis.
- If the calculated test statistic is less than the critical value(s) or the p-value is greater than the significance level, we fail to reject the null hypothesis.
G. To estimate the number of automobile thefts per year in a city with a population of about 50,000, we can use the theft rate per 10,000 people as a reference.
- The expected theft rate per 10,000 people can be calculated as (50,000 / 10,000) * theft rate = (5) * 44 = 220.
- Therefore, we would expect around 220 automobile thefts per year in a city with a population of about 50,000.
A. The hypothesis being tested would be whether or not the theft rate per 10,000 people has changed compared to the previous year.
B. The sample size of 36 locations can be considered relatively small.
C. The assumption that must be met before conducting the hypothesis test is that the data follows a normal distribution. Additionally, the sampling needs to be random and independent.
D. Since we're comparing rates per 10,000 people, the data would follow a Poisson distribution due to its discrete nature.
E. In this case, a two-tailed test would be selected because we are interested in determining if there is a significant change in either direction. A one-tailed test would only be used if we had specific evidence or expectation about which direction the change would occur.
F. To conduct the hypothesis test, we would use the following steps:
1. Set up the null hypothesis (H0): The theft rate per 10,000 people has not changed compared to the previous year.
2. Set up the alternative hypothesis (Ha): The theft rate per 10,000 people has changed compared to the previous year.
3. Calculate the test statistic using the data and the standard deviation provided. The test statistic formula would depend on the specific hypothesis test being used (e.g., z-test, t-test).
4. Determine the critical value corresponding to the desired significance level (e.g., 0.05, 0.01).
5. Compare the test statistic to the critical value. If the test statistic falls within the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
G. If your city's population is about 50,000, you can estimate the expected number of automobile thefts per year by multiplying the theft rate per 10,000 people by the number of population units:
Expected number of automobile thefts = Theft rate per 10,000 people * (Population / 10,000)
In this case, if the theft rate per 10,000 people is still 44, and the city's population is about 50,000, the estimated number of automobile thefts per year would be:
Expected number of automobile thefts = 44 * (50,000 / 10,000) = 220.