To find $P(\chi^2_{75} > 90.534)$, we are looking for the probability that a chi-square random variable with 75 degrees of freedom exceeds the value 90.534.

Steps:

1. Chi-Square Distribution: The chi-square distribution with 75 degrees of freedom is represented as $\chi^2_{75}$.

2. Use a Chi-Square Table or Calculator: To find $P(\chi^2_{75} > 90.534)$, you can either use a chi-square distribution table or a statistical calculator. This probability represents the area to the right of 90.534 under the chi-square distribution curve with 75 degrees of freedom.

3. Computing the Probability: We need the cumulative probability $P(\chi^2_{75} \leq 90.534)$ and then subtract it from 1 to find the tail probability.

   Using a calculator or statistical software, we find:

   $P(\chi^2_{75} \leq 90.534) \approx 0.9196$

   Therefore,

   $P(\chi^2_{75} > 90.534) = 1 - 0.9196 = 0.0804$

Final Answer:

$P(\chi^2_{75} > 90.534) \approx 0.0804$

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Would you like further clarification or assistance on this topic? Here are a few related questions:

1. How do you calculate probabilities for other distributions like normal or t-distribution?
2. What does the chi-square distribution represent in hypothesis testing?
3. How does the number of degrees of freedom affect the chi-square distribution?
4. What is the concept of critical values in chi-square tests?
5. How can you use software like R or Python to calculate chi-square probabilities?

Tip: When working with chi-square probabilities, always check if your software or table provides the cumulative distribution function (CDF) value directly or if you need to subtract from 1 to find the tail probability.