Questions: Consider a χ^2 distribution with 482 degrees of freedom. Find P(χ^2<483.8). Find P(χ^2>520.1). Find P(482.2<χ^2<543.8).

Consider a χ^2 distribution with 482 degrees of freedom.
Find P(χ^2<483.8).

Find P(χ^2>520.1).

Find P(482.2<χ^2<543.8).

Transcript text: Consider a $\chi^{2}$ distribution with 482 degrees of freedom. Find $P\left(\chi^{2}<483.8\right)$. Find $P\left(\chi^{2}>520.1\right)$. Find $P\left(482.2<\chi^{2}<543.8\right)$.

Solution

Solution Steps

To solve these problems, we will use the cumulative distribution function (CDF) of the chi-squared distribution. The CDF gives the probability that a chi-squared random variable is less than or equal to a certain value. For the first question, we directly use the CDF. For the second question, we use the complement rule, which involves subtracting the CDF from 1. For the third question, we find the difference between two CDF values to get the probability that the random variable falls within a specific range.

Step 1: Calculate \( P(\chi^{2} < 483.8) \)

Using the cumulative distribution function (CDF) of the chi-squared distribution with 482 degrees of freedom, we find: \[ P(\chi^{2} < 483.8) \approx 0.5316 \]

Step 2: Calculate \( P(\chi^{2} > 520.1) \)

To find this probability, we use the complement rule: \[ P(\chi^{2} > 520.1) = 1 - P(\chi^{2} < 520.1) \approx 1 - 0.8883 \approx 0.1117 \]

Step 3: Calculate \( P(482.2 < \chi^{2} < 543.8) \)

This probability is determined by finding the difference between the CDF values at 543.8 and 482.2: \[ P(482.2 < \chi^{2} < 543.8) = P(\chi^{2} < 543.8) - P(\chi^{2} < 482.2) \approx 0.4622 \]

Final Answer

\[ \boxed{ \begin{align_} P(\chi^{2} < 483.8) & \approx 0.5316 \\ P(\chi^{2} > 520.1) & \approx 0.1117 \\ P(482.2 < \chi^{2} < 543.8) & \approx 0.4622 \end{align_} } \]

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