A union of restaurant and foodservice workers would like to estimate the mean hourly wage, , of foodservice workers in the U.S. this year The mean hourly wage last year was $8.08, and there is good reason to believe that this year's value is greater than last year's.

The union decides to do a statistical test to see if the value has indeed increased. The union chooses a random sample of this year's wages, computes the mean of the sample to be $8.35, and computes the standard deviation of the sample to be $1.10. Based on this information, complete the parts below.(b)Suppose that the union decides to reject the null hypothesis. What sort of error might it be making? Type I or Type II

To address the question about hypothesis testing and the potential errors involved, let's break down the process and the concepts related to Type I and Type II errors.

### Hypothesis Testing

1. **Null Hypothesis ($ H_0 $)**: The mean hourly wage of foodservice workers this year is not greater than last year's mean wage, i.e., $ \mu \leq 8.08 $.
2. **Alternative Hypothesis ($ H_1 $)**: The mean hourly wage of foodservice workers this year is greater than last year's mean wage, i.e., $ \mu > 8.08 $.

Given the sample dataTo address the question about hypothesis testing and the potential errors involved, let's break down the process and the concepts related to Type I and Type II errors.

### Hypothesis Testing

1. **Null Hypothesis ($ H_0 $)**: The mean hourly wage of foodservice workers this year is not greater than last year's mean wage, i.e., $ \mu \leq 8.08 $.
2. **Alternative Hypothesis ($ H_1 $)**: The mean hourly wage of foodservice workers this year is greater than last year's mean wage, i.e., $ \mu > 8.08 $.

Given the sample 
- Sample mean ($ \bar{x} $) = $8.35
- Sample standard deviation ($ s $) = $1.10
- Sample size ($ n $)

The union conducts a statistical test (e.g., a one-sample t-test) to determine if the null hypothesis should be rejected in favor of the alternative hypothesis.

### Types of Errors in Hypothesis Testing

- **Type I Error (False Positive)**: This occurs when the null hypothesis is true, but it is incorrectly rejected. In this context, a Type I error would mean concluding that the mean hourly wage has increased when it actually has not.
- **Type II Error (False Negative)**: This occurs when the null hypothesis is false, but it is incorrectly failed to be rejected. In this context, a Type II error would mean concluding that the mean hourly wage has not increased when it actually has.

### Answer to the Given Question

**(b) Suppose that the union decides to reject the null hypothesis. What sort of error might it be making? Type I or Type II**

If the union decides to reject the null hypothesis ($ H_0 $), it is concluding that the mean hourly wage this year is greater than last year's mean wage of $8.08. If this conclusion is incorrect, the union is making a **Type I Error**.

### Summary
- **Type I Error**: Rejecting $ H_0 $ when $ H_0 $ is actually true. This is the error the union might be making if it decides to reject the null hypothesis and this conclusion is incorrect.

Would you like further details on how to calculate the p-value or the test statistic for this hypothesis test? If so, I can provide the specific steps and formulas.

### Related Questions

1. What is the significance level ($ \alpha $) in hypothesis testing?
2. How do you calculate the test statistic for a one-sample t-test?
3. What is the p-value, and how is it used in hypothesis testing?
4. How do you determine the critical value for a one-sample t-test?
5. What is the relationship between sample size and the power of a test?
6. How can you reduce the risk of making a Type I error?
7. What is the difference between a one-tailed and a two-tailed test?
8. How do confidence intervals relate to hypothesis testing?

#### Tip
Always check the assumptions of the statistical test you are using (e.g., normality, independence of observations) to ensure the validity of your test results.

Understand hypothesis testing, Type I and Type II errors, with an example of determining wage increases among foodservice workers in the U.S. Learn about rejecting null hypotheses and statistical significance. Sample mean: $8.35, Sample standard deviation: $1.10.

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