Number of Speeding Tickets

Oct 13, 2025 by miodeya.com 29 views

Question 2 of 6, Step 1 of 3

Introduction

In this lesson, we will learn how to test the fairness of law enforcement in its area by comparing the likelihood of women and men getting speeding tickets. This is a classic example of hypothesis testing for two populations. We will use the concept of hypothesis testing to determine whether there is a significant difference in the probability of women and men getting speeding tickets.

Background

The local citizens' group is concerned that women and men are not treated equally when it comes to speeding tickets. They want to know whether the probability of women getting speeding tickets is different from the probability of men getting speeding tickets. This is a classic example of a two-sample hypothesis test.

Hypothesis Testing for Two Populations

Hypothesis testing for two populations involves testing a hypothesis about the difference between the means of two populations. In this case, we want to test the hypothesis that the probability of women getting speeding tickets is different from the probability of men getting speeding tickets.

Null Hypothesis (H0)

The null hypothesis is a statement of no effect or no difference. In this case, the null hypothesis is:

H0: p1 = p2

where p1 is the probability of women getting speeding tickets and p2 is the probability of men getting speeding tickets.

Alternative Hypothesis (H1)

The alternative hypothesis is a statement of an effect or a difference. In this case, the alternative hypothesis is:

H1: p1 ≠ p2

where p1 is the probability of women getting speeding tickets and p2 is the probability of men getting speeding tickets.

Sample Data

The local citizens' group collects data on the number of speeding tickets given to women and men in a particular area. The data is as follows:

Gender	Number of Speeding Tickets
Women	250
Men	300

Calculating the Sample Proportions

To calculate the sample proportions, we need to divide the number of speeding tickets given to women and men by the total number of people in the sample.

Sample Proportion for Women (p̂1)

p̂1 = (Number of speeding tickets given to women) / (Total number of people in the sample) = 250 / 1000 = 0.25

Sample Proportion for Men (p̂2)

p̂2 = (Number of speeding tickets given to men) / (Total number of people in the sample) = 300 / 1000 = 0.30

Calculating the Standard Error

The standard error is a measure of the variability of the sample proportions. It is calculated as follows:

Standard Error (SE)

SE = √((p̂1 * (1 - p̂1)) / n1 + (p̂2 * (1 - p̂2)) / n2)

where n1 and n2 are the sample sizes for women and men, respectively.

SE = √((0.25 * (1 - 0.25)) / 500 + (0.30 * (1 - 0.30)) / 500) = √(0.019 + 0.021) = √0.04 = 0.2

Calculating the Z-Score

The Z-score is a measure of the number of standard errors that the sample proportions are away from the null hypothesis. It is calculated as follows:

Z-Score

Z = (p̂1 - p̂2) / SE

where p̂1 and p̂2 are the sample proportions for women and men, respectively.

Z = (0.25 - 0.30) / 0.2 = -0.05 / 0.2 = -0.25

Interpreting the Results

The Z-score is a measure of the number of standard errors that the sample proportions are away from the null hypothesis. In this case, the Z-score is -0.25, which means that the sample proportions are 0.25 standard errors away from the null hypothesis.

Conclusion

Based on the results, we can conclude that there is no significant difference in the probability of women and men getting speeding tickets. The Z-score is not significant, and we fail to reject the null hypothesis.

Limitations of the Study

There are several limitations of this study. Firstly, the sample size is relatively small, which may affect the accuracy of the results. Secondly, the data may not be representative of the entire population, which may affect the generalizability of the results.

Future Research Directions

There are several future research directions that can be explored. Firstly, a larger sample size can be used to increase the accuracy of the results. Secondly, a more representative sample can be used to increase the generalizability of the results. Finally, a more detailed analysis of the data can be performed to identify any underlying patterns or trends.

Conclusion

Frequently Asked Questions

Q1: What is hypothesis testing for two populations?

A1: Hypothesis testing for two populations is a statistical method used to compare the means of two populations to determine if there is a significant difference between them.

Q2: What is the null hypothesis in hypothesis testing for two populations?

A2: The null hypothesis is a statement of no effect or no difference. In hypothesis testing for two populations, the null hypothesis is typically that the means of the two populations are equal.

Q3: What is the alternative hypothesis in hypothesis testing for two populations?

A3: The alternative hypothesis is a statement of an effect or a difference. In hypothesis testing for two populations, the alternative hypothesis is typically that the means of the two populations are not equal.

Q4: What is the Z-score in hypothesis testing for two populations?

A4: The Z-score is a measure of the number of standard errors that the sample means are away from the null hypothesis. It is calculated as the difference between the sample means divided by the standard error.

Q5: How do I interpret the Z-score in hypothesis testing for two populations?

A5: The Z-score can be interpreted as follows:

If the Z-score is greater than 2, the null hypothesis can be rejected, and it is likely that the means of the two populations are different.
If the Z-score is less than -2, the null hypothesis can be rejected, and it is likely that the means of the two populations are different.
If the Z-score is between -2 and 2, the null hypothesis cannot be rejected, and it is likely that the means of the two populations are equal.

Q6: What are the limitations of hypothesis testing for two populations?

A6: The limitations of hypothesis testing for two populations include:

Small sample size: If the sample size is small, the results may not be accurate.
Non-representative sample: If the sample is not representative of the population, the results may not be generalizable.
Assumptions: Hypothesis testing for two populations assumes that the data is normally distributed and that the variances of the two populations are equal.

Q7: What are the future research directions for hypothesis testing for two populations?

A7: Future research directions for hypothesis testing for two populations include:

Increasing the sample size to increase the accuracy of the results.
Using a more representative sample to increase the generalizability of the results.
Performing a more detailed analysis of the data to identify any underlying patterns or trends.

Q8: How do I choose the significance level for hypothesis testing for two populations?

A8: The significance level is the probability of rejecting the null hypothesis when it is true. It is typically set at 0.05, but it can be set at any value between 0 and 1.

Q9: What is the difference between a one-tailed and a two-tailed test in hypothesis testing for two populations?

A9: A one-tailed test is used when the alternative hypothesis is directional, meaning that it specifies the direction of the effect. A two-tailed test is used when the alternative hypothesis is non-directional, meaning that it does not specify the direction of the effect.

Q10: How do I perform a hypothesis test for two populations in practice?

A10: To perform a hypothesis test for two populations in practice, follow these steps:

Formulate the null and alternative hypotheses.
Collect the data and calculate the sample means and standard errors.
Calculate the Z-score.
Interpret the Z-score and make a decision about the null hypothesis.
Report the results and discuss the implications of the findings.

By following these steps and understanding the concepts and limitations of hypothesis testing for two populations, you can perform a hypothesis test in practice and make informed decisions about the results.