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To test the hypothesis that the mean price of a cup of coffee is not equal to $4, let's go through the following steps:

1. **Set up the hypotheses**:
   - **Null Hypothesis (H0)**: The mean price of a cup of coffee is equal to $4.
   - **Alternative Hypothesis (H1)**: The mean price of a cup of coffee is not equal to $4.

   In formal terms:
   \[
   H_0: \mu = 4
   \]
   \[
   H_1: \mu \neq 4
   \]

2. **Significance Level**: The problem specifies a 10% significance level (α = 0.10).

3. **Given P-value**: The problem states that the p-value is 0.1069.

4. **Decision Rule**:
   - If the p-value is less than or equal to the significance level (0.10), we reject the null hypothesis.
   - If the p-value is greater than 0.10, we fail to reject the null hypothesis.

5. **Compare the P-value**:
   - Here, $ \text{p-value} = 0.1069 $.
   - Since 0.1069 > 0.10, we **fail to reject the null hypothesis**.

6. **Conclusion**:
   - Based on the test, there is **insufficient evidence to conclude that the mean price of a cup of coffee is different from $4** at the 10% significance level.

### Filling Out the Answers:

- **Hypotheses (Part a)**:
  - $ H_0 $: $ \mu = 4 $
  - $ H_1 $: $ \mu \neq 4 $

- **Decision (Part b)**:
  - **The correct decision is to fail to reject the null hypothesis**.

- **Summary**:
  - **The correct summary would be: "There is not sufficient evidence to conclude that the population mean price of a cup of coffee is not equal to $4."**

Would you like more details on any part of this process, or do you have any additional questions?

### Relative Questions
1. What is a significance level, and how is it used in hypothesis testing?
2. How does the p-value help us make decisions in hypothesis testing?
3. What does "failing to reject the null hypothesis" mean in practical terms?
4. Why do we set up hypotheses with equality in the null hypothesis?
5. How would the decision change if the significance level was lower (e.g., 5%)?

### Tip
Always check the p-value relative to the significance level to determine whether to reject the null hypothesis.

Test the claim that the mean price of a cup of coffee is not equal to $4 using a 10% level of significance. Assume that the coffee prices are normally distributed. Given a sample of 9 prices: [2.05, 3.65, 4.85, 3.4, 4.75, 3.2, 2.7, 3.25, 3.3], and a p-value of 0.1069, determine the appropriate hypotheses, decision, and summary.

Learn how to test a claim about the mean price of a cup of coffee using a two-tailed hypothesis test at a 10% significance level. Given a sample of prices and a p-value, determine the correct decision and conclusion.

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