Question 1176053
Let's solve this problem step-by-step.

**a. n = 10, S^2 <= 2.446**

1.  **Calculate the Chi-Squared Statistic (C):**
    * C = (n - 1) * S^2 / σ^2
    * C = (10 - 1) * 2.446 / 1.5
    * C = 9 * 2.446 / 1.5
    * C = 22.014 / 1.5
    * C = 14.676 (which rounds to 14.68 as given in the problem)

2.  **Degrees of Freedom (df):**
    * df = n - 1 = 10 - 1 = 9

3.  **Find the Probability Using the Chi-Squared Table:**
    * We need to find P(C ≤ 14.68) with 9 degrees of freedom.
    * Look up 14.68 in the Chi-Squared distribution table with 9 degrees of freedom.
    * Using a Chi-Squared table, we find that 14.68 falls between the values corresponding to cumulative probabilities of 0.10 and 0.05.
    * More specifically, for df = 9, the chi-square value for 0.10 is 14.684, and for 0.05 it is 16.919.
    * Since 14.676 is very close to 14.684, the probability P(C ≤ 14.68) is approximately 0.10.

**Therefore, P(S^2 ≤ 2.446) ≈ 0.10**

**b. n = 31, S^2 <= 0.83955**

1.  **Calculate the Chi-Squared Statistic (C):**
    * C = (n - 1) * S^2 / σ^2
    * C = (31 - 1) * 0.83955 / 1.5
    * C = 30 * 0.83955 / 1.5
    * C = 25.1865 / 1.5
    * C = 16.791

2.  **Degrees of Freedom (df):**
    * df = n - 1 = 31 - 1 = 30

3.  **Find the Probability Using the Chi-Squared Table or Scipy:**
    * We need to find P(C ≤ 16.791) with 30 degrees of freedom.

    * Using a Chi-Squared table, we find that the value 16.791 falls between the values corresponding to cumulative probabilities of 0.025 and 0.01.
    * More specifically, for df = 30, the chi-square value for 0.025 is 16.791.

    * Alternatively, we can use Python's Scipy library, as illustrated in the provided code, which calculates the cumulative distribution function (CDF) of the Chi-Squared distribution.

    * Based on the provided code, P(S^2 <= 0.83955) = 0.0250

**Therefore, P(S^2 ≤ 0.83955) ≈ 0.025**