5. opening a restaurant you are thinking about opening a restaurant and are searching for a good location…

5. opening a restaurant you are thinking about opening a restaurant and are searching for a good location. from your research, 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 site. based on the mean income of this sample, you will perform a test of $h_{0}:mu = $85,000$ versus $h_{a}:mu > $85,000$, where $mu$ is the true mean income in the population of people who live near the restaurant. describe a type i error and a type ii error in this setting.

5. opening a restaurant you are thinking about opening a restaurant and are searching for a good location. from your research, 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 site. based on the mean income of this sample, you will perform a test of $h_{0}:mu = $85,000$ versus $h_{a}:mu > $85,000$, where $mu$ is the true mean income in the population of people who live near the restaurant. describe a type i error and a type ii error in this setting.

Answer

Brief Explanations:

  • Type I error: Rejecting the null hypothesis (H_0:\mu = 85000) when it is actually true. In this context, it means concluding that the true mean income (\mu) of people living near the restaurant site is greater than ($85,000) ((H_a:\mu>85000)) when in reality (\mu = 85000). This could lead to opening an upscale restaurant in a location where the mean income is not actually high enough to support it.
  • Type II error: Failing to reject the null hypothesis (H_0:\mu = 85000) when the alternative hypothesis (H_a:\mu>85000) is true. Here, it would be not concluding that the mean income is greater than ($85,000) when it actually is. As a result, the restaurateur might miss out on a good location for the upscale restaurant.

Answer:

  • Type I error: Conclude (\mu > 85000) (open the restaurant) when (\mu=85000) (restaurant won't be supported by income).
  • Type II error: Fail to conclude (\mu > 85000) (don't open the restaurant) when (\mu > 85000) (miss a good location).