To solve this problem involving a chi-squared distribution with 10 degrees of freedom, we need to find specific probabilities using a chi-squared distribution table or a statistical calculator. Here’s a step-by-step solution for each part:

1. Find $P(\chi^2 < 9)$ for 10 degrees of freedom.

   • Using a chi-squared distribution table or calculator, look up the cumulative probability for $\chi^2 = 9$ with 10 degrees of freedom.
   • Based on common tables, $P(\chi^2 < 9) \approx 0.5$.

2. Find $P(\chi^2 > 9.7)$ for 10 degrees of freedom.

   • This is the complement of the cumulative probability for $\chi^2 \leq 9.7$.
   • Using a chi-squared distribution table, $P(\chi^2 < 9.7) \approx 0.55$, so $P(\chi^2 > 9.7) = 1 - 0.55 = 0.45$.

3. Find $P(9.4 < \chi^2 < 14.2)$ for 10 degrees of freedom.

   • This is the difference between two cumulative probabilities: $P(\chi^2 < 14.2)$ and $P(\chi^2 < 9.4)$.
   • From the table:
     • $P(\chi^2 < 14.2) \approx 0.75$.
     • $P(\chi^2 < 9.4) \approx 0.53$.
   • So, $P(9.4 < \chi^2 < 14.2) = 0.75 - 0.53 = 0.22$.

To confirm these values or get more precise results, you could use statistical software or an online chi-squared calculator with 10 degrees of freedom.

Would you like further clarification on any of these steps or a detailed explanation of the chi-squared distribution?

Follow-up Questions

1. What are degrees of freedom in a chi-squared distribution, and why are they important?
2. How can the cumulative distribution function (CDF) be used to find probabilities in chi-squared distributions?
3. What are some practical applications of the chi-squared distribution in real-world scenarios?
4. How does the chi-squared distribution differ from a normal distribution?
5. How can statistical software like R or Python’s SciPy library be used to compute chi-squared probabilities?

Tip

When working with the chi-squared distribution, remember that probabilities can often be approximated using tables, but for higher precision, statistical calculators or software are highly useful.