https://file.aiquickdraw.com/mathbot/file/17302201730426bexcz0f.jpg

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?

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**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).

Find the p-value if left-sided test, t-distribution, n = 45, Test Statistic = -2.5

Learn how to calculate the p-value for a left-sided t-test with a t-distribution, sample size of 45, and a test statistic of -2.5. This solution includes the use of degrees of freedom and step-by-step p-value calculation for hypothesis testing.

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