Opening a restaurant You are thinking about opening a...

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 “standard” significance levels for your significance test, would you choose A 0.01, 0.05, or 0.10? Justify your choice.

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 “standard” significance levels for your significance test, would you choose A 0.01, 0.05, or 0.10? Justify your choice.

Key Concepts

Type I and Type II Errors

Type I and Type II errors are the two possible mistakes in the context of hypothesis testing. A Type I error occurs when the test incorrectly rejects a true null hypothesis (a false positive), potentially leading to unnecessary changes or actions. Conversely, a Type II error happens when the test fails to reject a false null hypothesis (a false negative), which might result in missing an important effect or difference that actually exists. Understanding and managing these errors is critical for making informed decisions based on statistical analysis.

Significance Level

The significance level, often denoted by alpha (?), is the threshold set before testing that defines the maximum probability of committing a Type I error. It influences the test's stringency, with lower levels requiring stronger evidence against the null hypothesis. Choosing an appropriate significance level balances the risks of false positives and false negatives according to the context and consequences of decision making.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents the default or established belief about a population parameter, whereas the alternative hypothesis is the claim that opposes the null and is supported by evidence provided by a sample. Setting these hypotheses correctly is essential for determining whether the data collected provides sufficient evidence to support a new claim or change in belief regarding the parameter.

Transcript

00:01 You are thinking about opening a restaurant and are searching for a good location.

00:05 You know that the mean income of those living near the restaurant must be $85 ,000 or over to support the type of upscale restaurant you wish to have.

00:14 So in this case, we're going to let our mew, our mean, represent the mean income of residents near the restaurant that you're trying to open.

00:25 We're going to say that in our significance test, the nol hypothesis, will be that, that mue is equal to that average income of $85 ,000.

00:38 And we're gonna say that our alternative hypothesis, our alternative hypothesis is gonna be that mew is greater than $85 ,000.

00:50 So this is gonna be a one -sided alternative hypothesis because we want people to be able to afford our upscale restaurant.

01:01 Now, you can run your significance test and you get your p value and compare it to an alpha and sometimes a type one error can happen.

01:11 This is where we reject the null hypothesis when it is true.

01:15 In this case, a type 1 error would represent that we open up the restaurant in a location where the residents will not be able to support it or, go to it because they don't make as much money as we had thought they did.

01:42 So a type 1 error in this case would that you open the restaurant.

01:47 Now a type 2 error is when we fail to reject the null hypothesis when the alternative is true.

01:55 And in this case, if we were to make a type 2 error, this means that we do not open the restaurant in a location where the residents would have been able to afford it...

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