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?

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.