1. **Problem statement:** We need to find probabilities involving the chi-square ($x^2$) distribution with given degrees of freedom (d.f.). The chi-square distribution is used in statistics to describe the distribution of a sum of squared standard normal variables. 2. **Formula and rules:** The probability for chi-square values is found using the chi-square cumulative distribution function (CDF). For example, $P(x^2 > a) = 1 - P(x^2 \leq a)$ and $P(b < x^2 < c) = P(x^2 \leq c) - P(x^2 \leq b)$. 3. **Step-by-step solutions:** (a) $P(x^2 > 31.33)$ with d.f. = 18: - Find $P(x^2 \leq 31.33)$ from chi-square tables or software. - Then $P(x^2 > 31.33) = 1 - P(x^2 \leq 31.33)$. (b) $P(x^2 < 1.15)$ with d.f. = 5: - Directly find $P(x^2 \leq 1.15)$ from chi-square tables or software. (c) $P(3.24 < x^2 < 18.31)$ with d.f. = 10: - Find $P(x^2 \leq 18.31)$ and $P(x^2 \leq 3.24)$. - Then subtract: $P = P(x^2 \leq 18.31) - P(x^2 \leq 3.24)$. (d) $P(3.49 < x^2 < 20.09)$ with d.f. = 8: - Find $P(x^2 \leq 20.09)$ and $P(x^2 \leq 3.49)$. - Then subtract: $P = P(x^2 \leq 20.09) - P(x^2 \leq 3.49)$. 4. **Explanation in simple terms:** - Think of $x^2$ as a number that can be bigger or smaller. - We want to know how likely it is to be bigger or smaller than some numbers. - We use special tables or calculators that tell us these chances. - For "greater than" problems, we subtract from 1 because tables give "less than" probabilities. - For "between" problems, we find the chance up to the bigger number and subtract the chance up to the smaller number. 5. **Final answers (approximate using chi-square CDF):** (a) $P(x^2 > 31.33) \approx 0.028$ (b) $P(x^2 < 1.15) \approx 0.050$ (c) $P(3.24 < x^2 < 18.31) \approx 0.850$ (d) $P(3.49 < x^2 < 20.09) \approx 0.870$