Question

You are thinking about opening a restaurant and are searching for a good location. From the 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.8

(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 $\alpha$=0.01, 0.05, or 0.10? Justify your choice.

Short answer

a. $H_0:\mu =\mathrm{\$}85,000$-for null hypothesis and $H_a:\mu >\mathrm{\$}85,000$- for alternative hypothesis

b. Type I error- mean income = $\$85000$, but tests show it is more. Type II error - mean income greater than$\$85000$but tests shows it is equal

c.$\alpha =0.01$

Step-by-step solution

introduction

The null hypothesis is an overall explanation that expresses that there is no connection between two peculiarities viable or that there is no relationship between two gatherings. An alternative hypothesis is an explanation that depicts that there is a connection between two chosen factors in a study

explanation part (a)

Mean income is $\$85000$

calculating the null and alternative hypotheses,

$H_0:\mu =\mathrm{\$}85,000$- for null hypothesis

$H_a:\mu >\mathrm{\$}85,000$- for alternative hypothesis

hence mean income is equal to $\$85000$and greater than it as per the null and alternative hypotheses respectively.

explanation part (b)

Mean income is $\$85000$

Type I error- $H_0$is dismissed when the null hypothesis is valid

The mean income is equivalent to $\$85000$however the test indicates that the mean income is more.

Subsequently, the result is that the café will open in that location since the mean income is more than the required income, yet in reality, it is not a decent location for the eatery.

Type II error- when the null hypothesis is valid then rejection of $H_0$additionally failed.

The mean income is more than $\$85000$while the test indicates that the mean income is equivalent.

Consequently, the outcome is that the eatery will not open in that location since the mean income is more modest than the required income, yet in reality, it is a decent location for the café.

explanation part (c)

To open a restaurant requires a huge investment. Consequently, there is a possibility of making incorrect decisions which can cost a huge amount of cash.

Consequently, a more modest significance level $\alpha =0.01$will be smarter to minimize the probability of making some incorrect decision.