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To assist with this hypothesis testing problem, let’s go through the steps in detail based on what’s shown in the image.

### Problem Summary
Payton wants to determine if the average sales of McDonald's restaurants in Arkansas are different from the worldwide average of $3.7 million. The Arkansas sample consists of 44 restaurants with given sales data, and a hypothesis test will be performed at a 9% significance level ($ \alpha = 0.09 $).

### Step 1: State the Null and Alternative Hypotheses
- **Null Hypothesis ($ H_0 $)**: The average sales in Arkansas are equal to the worldwide average, $ \mu = 3.7 $ million.
- **Alternative Hypothesis ($ H_a $)**: The average sales in Arkansas are different from the worldwide average, $ \mu \neq 3.7 $ million.
- This is a **two-tailed** test because we are looking for any difference, not just an increase or decrease.

### Step 2: Sampling Distribution
1. **Distribution Type**: Since the sample size is 44 (which is greater than 30), by the Central Limit Theorem, the sampling distribution can be approximated as a **normal distribution**.
2. **Mean of Sampling Distribution**: The mean is the population mean, which is **3.7 million** according to the null hypothesis.
3. **Standard Error of the Mean (SEM)**: This is calculated using the formula:
   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}}
   \]
   where $ \sigma = 0.5 $ (the standard deviation of the population) and $ n = 44 $ (the sample size).

   \[
   \text{SEM} = \frac{0.5}{\sqrt{44}} \approx 0.0754
   \]

Therefore, the sampling distribution is:
- **Normal distribution** with a mean of **3.7** and a **standard error of approximately 0.0754**.

Let me know if you'd like further explanation or help with additional steps like calculating the test statistic or finding the critical value.

### Follow-up Questions
1. How do we determine the critical values for a two-tailed test at a 9% significance level?
2. What would be the next steps if we were to calculate the test statistic?
3. Why is the Central Limit Theorem important in this hypothesis test?
4. What would change if the sample size were below 30?
5. How do we interpret the results if the p-value is less than 0.09?

### Tip
In hypothesis testing, always verify assumptions (like sample size and normality) to ensure the validity of the test results.

Payton wants to know if the average sales generated by McDonald's restaurants in Arkansas is different from the worldwide average of $3.7 million with a standard deviation of $0.5 million. She surveys 44 restaurants in Arkansas and finds the following data (in millions of dollars): 3.6, 4.3, 3.9, 3.2, etc. Perform a hypothesis test using a 9% level of significance.

Learn how to perform a hypothesis test to check if McDonald's sales in Arkansas differ from the global average of $3.7 million. This example includes calculating standard error, using the Central Limit Theorem, and setting up a two-tailed test at a 9% significance level. Suitable for undergraduate statistics courses.

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