https://file.aiquickdraw.com/mathbot/file/17334412815000nlv3e0w.jpg

From the given image, I see a hypothesis testing problem focused on comparing a population mean to a hypothesized value using a t-test. Here's a step-by-step interpretation:

### Analysis of the Problem:

1. **Test Type**:  
   A one-sample **t-test for a population mean** is used since the population standard deviation is not mentioned, and the sample size is being compared against a specific hypothesized mean.

2. **Null and Alternative Hypotheses**:  
   - Null Hypothesis (\(H_0\)): \( \mu = 38 \)  
     This suggests that the population mean is equal to 38 days.  
   - Alternative Hypothesis (\(H_1\)): \( \mu < 38 \)  
     This is a one-tailed test to determine if the population mean is significantly lower than 38 days.

3. **Test Statistic Calculation**:  
   The t-statistic value provided is \( t = -2.025 \). This measures how many standard errors the sample mean deviates from the hypothesized mean.

4. **P-value**:  
   The provided p-value is \( p = 0.035 \), which indicates the probability of observing the test results (or more extreme) if the null hypothesis is true.

5. **Significance Level (\( \alpha \))**:  
   The comparison threshold is \( \alpha = 0.05 \). Since \( p \leq \alpha \), we reject the null hypothesis.

### Conclusion:
The correct interpretation is:
- **Option 2**:  
  "The data suggest the population mean is significantly lower than 38 at \( \alpha = 0.05 \), so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is less than 38."

Would you like a breakdown of how the t-statistic and p-value are derived? Or do you have further questions?  

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### Related Questions:
1. What formula is used to calculate the t-statistic in hypothesis testing?  
2. Why do we compare the p-value to \( \alpha \) in significance testing?  
3. How does a one-tailed test differ from a two-tailed test in hypothesis testing?  
4. What are the assumptions behind using a t-test for a population mean?  
5. How is the degree of freedom determined in a t-test?

### Tip:
Always double-check whether the test is one-tailed or two-tailed when setting up the hypotheses to avoid incorrect conclusions.

For this study, we should use a t-test for a population mean. Determine the null and alternative hypotheses, compute the test statistic, and interpret the results based on the given data (t = -2.025, p = 0.035, α = 0.05).

Explore hypothesis testing using a one-sample t-test for a population mean. This guide includes determining null and alternative hypotheses, calculating the t-statistic (t = -2.025), interpreting the p-value (p = 0.035), and making statistical conclusions at a 0.05 significance level. Ideal for college statistics students.

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