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To find the P-value for this left-tailed chi-square hypothesis test, we can follow these steps:

1. **Determine Degrees of Freedom**: The degrees of freedom for this chi-square distribution is given as \( df = 51 \).
2. **Given Chi-Square Value**: The test statistic provided is \( \chi^2_{51} = 35.548 \).
3. **Left-Tailed Test**: Since this is a left-tailed test, we are interested in the probability \( P(\chi^2 \leq 35.548) \) for a chi-square distribution with 51 degrees of freedom.

Using a chi-square distribution table or calculator, we can find the cumulative probability up to the test statistic value for a chi-square distribution with 51 degrees of freedom.

The approximate P-value for \( \chi^2 = 35.548 \) with \( df = 51 \) is:

\[
P \approx 0.9652
\]

### Answer:
**P-Value = 0.9652**

Would you like a more detailed breakdown or have any questions?

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### Related Questions:
1. How is the P-value interpreted in hypothesis testing?
2. Why is it important to know if the test is one-tailed or two-tailed?
3. What changes in interpretation if the test were right-tailed instead of left-tailed?
4. How do degrees of freedom impact the shape of the chi-square distribution?
5. What conditions must be met to use a chi-square test?

### Tip:
When performing a chi-square test, always verify if it's one-tailed or two-tailed, as this affects how you calculate the P-value.

A sample of 52 values is taken from a population that is normally distributed. You are doing a left-tailed test and the test statistic is χ²₅₁ = 35.548. Find the P-value for this hypothesis test.

Calculate the P-value for a left-tailed chi-square test with 51 degrees of freedom and a test statistic of χ² = 35.548. This problem demonstrates key concepts in hypothesis testing using chi-square distribution. Suitable for college-level statistics students.

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