The problem given is a hypothesis test question asking for the p-value. Here is a breakdown of the relevant information extracted from the image:

• Test Type: Left-sided test
• Distribution: t-distribution
• Sample Size $(n)$: 45
• Test Statistic: $-2.5$

Step-by-Step Solution

To find the p-value for a left-tailed test with a $t$-distribution:

1. Degrees of Freedom (df): For a t-test, degrees of freedom is $n - 1$. $df = 45 - 1 = 44$

2. Using the Test Statistic: Since it's a left-sided test, the p-value is the probability that a t-distributed random variable with 44 degrees of freedom is less than $-2.5$.

3. Finding the p-value: You would typically use a t-distribution table or statistical software to find this value.

   For a $t$-score of $-2.5$ with 44 degrees of freedom, the approximate p-value is 0.007 (as given in the image).

Conclusion

If this p-value is compared against a significance level $\alpha$, such as $\alpha = 0.05$, you would reject the null hypothesis because $0.007 < 0.05$.

Would you like further details on how to calculate or interpret this result?

Further Questions:

1. What does the p-value represent in the context of hypothesis testing?
2. How does the choice between a one-tailed and two-tailed test affect the p-value?
3. Why do we use degrees of freedom in the t-distribution?
4. What conditions must be met to use a t-distribution instead of a z-distribution?
5. How would the interpretation change if this were a two-tailed test?

Tip: For hypothesis testing, always verify the type of test (one-tailed or two-tailed) and check if the conditions match the assumptions of the test being used (like normality for small samples in t-tests).