In A Small Town Of 5,832 People, The Mayor Wants To Determine If There Is A Difference In The Proportion Of Voters Ages 18-30 Who Would Support An Increase In The Food Tax, And The Proportion Of Voters Ages 31-40 Who Would Support An Increase In The

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Introduction

In a small town of 5,832 people, the mayor wants to determine if there is a difference in the proportion of voters ages 18-30 who would support an increase in the food tax, and the proportion of voters ages 31-40 who would support an increase in the same tax. This is a classic example of a hypothesis testing problem, where we want to test whether there is a significant difference between two proportions. In this article, we will outline the steps involved in conducting a hypothesis test for proportions and provide a case study to illustrate the process.

Background

The mayor of the small town has proposed an increase in the food tax to fund various community projects. However, the proposal has been met with resistance from some voters, particularly those in the 18-30 age group. The mayor wants to know whether there is a significant difference in the proportion of voters in this age group who support the increase in the food tax, compared to voters in the 31-40 age group. This information will help the mayor to better understand the opinions of the voters and make informed decisions about the proposal.

Hypothesis Testing for Proportions

Hypothesis testing for proportions is a statistical technique used to determine whether there is a significant difference between two or more proportions. The process involves the following steps:

  1. Formulate the null and alternative hypotheses: The null hypothesis (H0) is a statement of no effect or no difference, while the alternative hypothesis (H1) is a statement of an effect or a difference. In this case, the null hypothesis is that there is no difference in the proportion of voters ages 18-30 who support the increase in the food tax, and the proportion of voters ages 31-40 who support the same tax. The alternative hypothesis is that there is a significant difference between the two proportions.
  2. Choose a significance level: The significance level (α) is the maximum probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.05 and 0.01.
  3. Select a test statistic: The test statistic is a numerical value that is used to determine whether the null hypothesis can be rejected. In this case, we will use the z-test statistic, which is calculated as follows:

z = (p1 - p2) / sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))

where p1 and p2 are the proportions of voters ages 18-30 and 31-40 who support the increase in the food tax, respectively, and n1 and n2 are the sample sizes for each age group.

  1. Determine the critical region: The critical region is the range of values for the test statistic that would lead to the rejection of the null hypothesis. This is determined by the significance level and the test statistic.
  2. Make a decision: If the test statistic falls within the critical region, the null hypothesis is rejected, and it is concluded that there is a significant difference between the two proportions.

Case Study

Let's assume that a survey of 1,000 voters in the small town was conducted, with 500 voters in the 18-30 age group and 500 voters in the 31-40 age group. The survey results showed that 60% of voters in the 18-30 age group supported the increase in the food tax, while 40% of voters in the 31-40 age group supported the same tax. We want to determine whether there is a significant difference between the two proportions.

First, we need to formulate the null and alternative hypotheses:

H0: p1 = p2 (there is no difference in the proportion of voters ages 18-30 who support the increase in the food tax, and the proportion of voters ages 31-40 who support the same tax)

H1: p1 ≠ p2 (there is a significant difference between the two proportions)

Next, we need to choose a significance level. Let's assume that we choose a significance level of 0.05.

Now, we need to select a test statistic. We will use the z-test statistic, which is calculated as follows:

z = (0.6 - 0.4) / sqrt((0.6 * (1 - 0.6) / 500) + (0.4 * (1 - 0.4) / 500))

z = 0.2 / sqrt(0.00072 + 0.00032)

z = 0.2 / sqrt(0.00104)

z = 0.2 / 0.032

z = 6.25

Now, we need to determine the critical region. For a significance level of 0.05, the critical region is z > 1.96 or z < -1.96.

Finally, we need to make a decision. Since the test statistic (z = 6.25) falls within the critical region (z > 1.96), we reject the null hypothesis and conclude that there is a significant difference between the proportion of voters ages 18-30 who support the increase in the food tax, and the proportion of voters ages 31-40 who support the same tax.

Conclusion

In this article, we outlined the steps involved in conducting a hypothesis test for proportions and provided a case study to illustrate the process. We demonstrated how to formulate the null and alternative hypotheses, choose a significance level, select a test statistic, determine the critical region, and make a decision. The case study showed that there is a significant difference between the proportion of voters ages 18-30 who support the increase in the food tax, and the proportion of voters ages 31-40 who support the same tax. This information can be used by the mayor to better understand the opinions of the voters and make informed decisions about the proposal.

Limitations

There are several limitations to this study. First, the sample size was relatively small, which may have affected the accuracy of the results. Second, the survey was conducted in a small town, which may not be representative of the larger population. Finally, the survey results may have been influenced by various factors, such as the way the questions were phrased and the response rate.

Future Research

Future research could involve conducting a larger survey to increase the accuracy of the results. Additionally, the survey could be conducted in a larger population to increase the generalizability of the results. Finally, the survey could be designed to control for various factors that may have influenced the results.

References

Note: The references provided are for illustrative purposes only and are not actual references used in the case study.

Introduction

Hypothesis testing for proportions is a statistical technique used to determine whether there is a significant difference between two or more proportions. In this article, we will answer some frequently asked questions about hypothesis testing for proportions.

Q: What is the purpose of hypothesis testing for proportions?

A: The purpose of hypothesis testing for proportions is to determine whether there is a significant difference between two or more proportions. This can be used to make informed decisions about a particular issue or problem.

Q: What are the steps involved in hypothesis testing for proportions?

A: The steps involved in hypothesis testing for proportions are:

  1. Formulate the null and alternative hypotheses
  2. Choose a significance level
  3. Select a test statistic
  4. Determine the critical region
  5. Make a decision

Q: What is the null hypothesis in hypothesis testing for proportions?

A: The null hypothesis in hypothesis testing for proportions is a statement of no effect or no difference. For example, "there is no difference in the proportion of voters ages 18-30 who support the increase in the food tax, and the proportion of voters ages 31-40 who support the same tax."

Q: What is the alternative hypothesis in hypothesis testing for proportions?

A: The alternative hypothesis in hypothesis testing for proportions is a statement of an effect or a difference. For example, "there is a significant difference between the proportion of voters ages 18-30 who support the increase in the food tax, and the proportion of voters ages 31-40 who support the same tax."

Q: What is the significance level in hypothesis testing for proportions?

A: The significance level in hypothesis testing for proportions is the maximum probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.05 and 0.01.

Q: What is the test statistic in hypothesis testing for proportions?

A: The test statistic in hypothesis testing for proportions is a numerical value that is used to determine whether the null hypothesis can be rejected. The most commonly used test statistic is the z-test statistic.

Q: What is the critical region in hypothesis testing for proportions?

A: The critical region in hypothesis testing for proportions is the range of values for the test statistic that would lead to the rejection of the null hypothesis.

Q: How do I determine the critical region in hypothesis testing for proportions?

A: The critical region in hypothesis testing for proportions is determined by the significance level and the test statistic. For example, if the significance level is 0.05 and the test statistic is a z-test statistic, the critical region would be z > 1.96 or z < -1.96.

Q: What is the decision in hypothesis testing for proportions?

A: The decision in hypothesis testing for proportions is whether to reject or not reject the null hypothesis. If the test statistic falls within the critical region, the null hypothesis is rejected, and it is concluded that there is a significant difference between the two proportions.

Q: What are the limitations of hypothesis testing for proportions?

A: The limitations of hypothesis testing for proportions include:

  • Small sample size
  • Limited generalizability
  • Potential for bias in the data

Q: What are some common applications of hypothesis testing for proportions?

A: Some common applications of hypothesis testing for proportions include:

  • Determining whether there is a significant difference between the proportion of voters who support a particular issue or policy
  • Evaluating the effectiveness of a particular treatment or intervention
  • Comparing the proportion of individuals who exhibit a particular behavior or characteristic

Conclusion

Hypothesis testing for proportions is a statistical technique used to determine whether there is a significant difference between two or more proportions. In this article, we have answered some frequently asked questions about hypothesis testing for proportions, including the purpose, steps, null and alternative hypotheses, significance level, test statistic, critical region, decision, limitations, and common applications.

References