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Opening a restaurant You are thinking about opening a restaurant and are searching for a good location. From research you have done, you know that the mean income of those living near the restaurant must be over $85,000 to support the type of upscale restaurant you wish to open. You decide to take a simple random sample of 50 people living near one potential location. Based on the mean income of this sample, you will decide whether to open a restaurant there. (a) State appropriate null and alternative hypotheses. Be sure to define your parameter. (b) Describe a Type I and a Type II error, and explain the consequences of each. (c) If you had to choose one of the 鈥渟tandard鈥� significance levels for your significance test, would you choose A 0.01, 0.05, or 0.10? Justify your choice.

Step by step solution

01

Define the Parameter and State Hypotheses

The parameter of interest is the mean income of people living near the potential restaurant location. Let \( \mu \) represent this mean income.- **Null Hypothesis (\( H_0 \)):** \( \mu = 85,000 \); the mean income is \(85,000.- **Alternative Hypothesis (\( H_a \)):** \( \mu > 85,000 \); the mean income is greater than \)85,000.

02

Describe Type I Error and Its Consequence

A Type I error occurs when the null hypothesis is true, but we reject it. In this context, it means concluding that the mean income is greater than $85,000, and deciding to open the restaurant, when in fact the mean income is $85,000. **Consequence:** This could lead to financial loss since the assumption that the local population can support an upscale restaurant might be incorrect.

03

Describe Type II Error and Its Consequence

A Type II error occurs when the null hypothesis is false, but we fail to reject it. In this context, it means deciding not to open the restaurant because we believe the mean income is $85,000 or less, when in fact it is greater. **Consequence:** Missing an opportunity to successfully open a profitable restaurant in an area where the clientele can afford it.

04

Choose the Significance Level

The choice of significance level (\( \alpha \)) affects the probabilities of Type I and II errors.Given the upscale nature of the restaurant and the potential financial risk, it's crucial to avoid a Type I error. However, missing an opportunity also has costs. A balance can be found at a moderate significance level.**Chosen Level:** A significance level of 0.05 balances the risks of both Type I and Type II errors, making it a common choice in business decisions where neither error consequence dramatically outweighs the other.

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Key Concepts

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

Null Hypothesis

When conducting hypothesis testing, it's essential to understand the null hypothesis. In this scenario, the null hypothesis is a statement about the mean income of people living near the potential restaurant location. The null hypothesis (denoted as \( H_0 \)) is typically a statement of no effect or no difference. Here, it is defined as \( \mu = 85,000 \), asserting that the mean income is precisely \(85,000.

• The null hypothesis is what you test against. It's the default position that there is no relationship between two measured phenomena.
• Rejecting the null hypothesis can lead you to take action, such as deciding to open a restaurant if the mean is significantly greater than \)85,000.

Establishing \( H_0 \) is critical because it sets the stage for statistical testing, allowing you to quantify the uncertainty in your data and make an informed decision.

Alternative Hypothesis

Once you have a null hypothesis, you need an alternative hypothesis to test against it. The alternative hypothesis (denoted as \( H_a \)) is what you might conclude if there is statistical evidence against the null hypothesis.

In this case, the alternative hypothesis is \( \mu > 85,000 \), suggesting that the mean income of people is indeed greater than $85,000. The alternative hypothesis reflects the business's hope and expectation because supporting it could mean moving forward with the restaurant opening at the location.

• It often represents the idea or theory you think might be true.
• The task of hypothesis testing is to assess the strength of evidence against \( H_0 \) in favor of \( H_a \).

Choosing \( H_a \) with "greater than" indicates a one-tailed test, focusing only on the possibility of the income being more than the threshold required for the restaurant.

Type I Error

A Type I error is also known as a "false positive." It occurs when you reject a true null hypothesis. In the context of the restaurant scenario, this error means concluding that the area's mean income is greater than \(85,000, leading to the decision to open the restaurant, while the true mean income is actually \)85,000 or less.

• Type I errors can have significant consequences, such as financial loss from investing in an unsustainable business location.
• The probability of committing this error is defined by the significance level \( \alpha \).

Choosing a lower significance level reduces the chance of making a Type I error, but it can increase the possibility of a Type II error, reflecting a trade-off between different types of risks.

Type II Error

On the other hand, a Type II error is referred to as a "false negative." It occurs when you fail to reject a false null hypothesis. In the restaurant example, a Type II error means you decide not to open the restaurant because you believe the mean income is $85,000 or less, while it is actually more.

• This could result in missed business opportunities, where a potentially profitable restaurant is not opened.
• The probability of making a Type II error is denoted by \( \beta \), and it is inversely related to the test's power.

Balancing the risks of Type I and Type II errors in business decisions involves selecting an appropriate significance level, which guides you in making decisions under uncertainty.