91Ó°ÊÓ

In 2015 , the U.S. Census Bureau reported that \(62.2 \%\) of American families owned their homes the lowest rate in 20 years. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers to encourage people to become homeowners. They decide to adopt the plan on a 2 -year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. a. In words, what will their hypotheses be? b. What would a Type I error be? c. What would a Type II error be? d. For each type of error, tell who would be harmed. e. What would the power of the test represent in this context?

Step by step solution

01

- Hypothesis Formation

The Null Hypothesis (H0): The rate of home ownership has not increased or has decreased. The Alternative Hypothesis (H1): The rate of home ownership has increased.

02

- Define Type I Error

A Type I error would occur if the city council concludes that the rate of home ownership has increased when it really has not.

03

- Define Type II Error

A Type II error would occur if the city council concludes that the rate of home ownership has not increased when it really has.

04

- Identify Harm from Each Type of Error

In case of a Type I error, the city council and the tax payers would be harmed as tax revenue would be reduced without achieving increased home ownership. In case of a Type II error, potential home owners would be harmed as they would miss the opportunity of getting tax breaks to own a home.

05

- Power of Test

The power of the test here represents the council's ability to correctly conclude that the rate of home ownership is increasing when it actually is.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I error

In hypothesis testing, a Type I error refers to the situation where the test wrongly rejects the null hypothesis when it is actually true. In simple terms, it's like sounding a false alarm. In the context of the city's tax break plan, a Type I error would mean that the city council decides the home ownership rate has increased due to the tax breaks, when in fact, it has not. This type of error could lead to continued loss of tax revenue without any real benefit, as the city would continue offering tax breaks without achieving the intended increase in home ownership. Here, the taxpayers and the council itself would be harmed by the financial implications of this incorrect decision. It's crucial for the council to be cautious and not prematurely conclude success based on flawed data.

Type II error

A Type II error happens when the test fails to reject the null hypothesis when the alternative hypothesis is actually true. In simpler terms, it's like missing out on a true effect or a success. For the city council, a Type II error would occur if they conclude that the home ownership rate hasn't increased when it truly has. The consequence of this error is that potential homeowners who could benefit from the tax breaks are denied this support. This means the opportunity to promote home ownership and potentially increase the city's growth and development is lost. The council should ensure their analysis methods are robust to minimize the chances of overlooking real, positive outcomes.

Statistical power

Statistical power is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. In other words, power is the test's ability to detect a real effect. For the city council's plan, the power of the test is a measure of their ability to correctly claim that home ownership rates have increased due to the tax breaks when they really have. High statistical power means that the city can be confident in its finding of increased home ownership and decide on continuing the tax breaks based on solid evidence. A powerful test usually results from a large sample size and effectively designed data collection methods, helping ensure the council's decisions are based on strong and reliable data.

Null hypothesis

The null hypothesis is a statement that assumes no effect or no change in a certain condition. It acts as a baseline for statistical testing. In this scenario, the null hypothesis for the city council would be that the rate of home ownership has not increased, or perhaps has decreased, even with the tax breaks. It's important because it provides a standard that the data collected must overcome to show a real impact of the tax breaks. The whole aim of the hypothesis test is to determine if the data is strong enough to reject this null hypothesis and suggest a positive change due to the council’s initiatives.

Alternative hypothesis

The alternative hypothesis is the statement that there is an effect or a change, opposing the null hypothesis. It proposes the scenario the researchers hope to find evidence for. For the city council, the alternative hypothesis is that the home ownership rate has actually increased due to the tax incentives offered. This hypothesis represents the desired outcome, showing that the tax break is effective. Establishing the alternative hypothesis gives the council a clear goal to test against the null hypothesis. Successful rejection of the null hypothesis in favor of the alternative could lead to the continuation of the tax benefits, supporting new homeowners and potentially boosting the city's economic growth. The alternative hypothesis guides the entire testing process by setting the desired outcome that the council wants to prove.