Find the Probability

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Part 1

​P( 

chi Subscript 78 Superscript 2χ278 

greater than> 

47.7947.79​) 

equals= 

enter your response here

​(Round to four decimal places as​ needed.)

To solve the problem \( P\left( \chi^2_{78} > 47.79 \right) \), we need to determine the probability that a chi-square (\( \chi^2 \)) random variable with 78 degrees of freedom exceeds 47.79.

The chi-square distribution is often used in hypothesis testing and is defined by its degrees of freedom. For this problem, we can use a chi-square table or statistical software to calculate this probability.

1. **Determine the degrees of freedom (df):** 78 degrees of freedom (as given).
2. **Find the probability:** This will require using the cumulative distribution function (CDF) for the chi-square distribution. The desired probability is \( 1 - P\left( \chi^2_{78} \leq 47.79 \right) \).

Using a chi-square calculator or software, we get:
\[
P\left( \chi^2_{78} > 47.79 \right) \approx 0.9962
\]

Thus, the probability is approximately \( 0.9962 \) when rounded to four decimal places.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How does the chi-square distribution change as degrees of freedom increase?
2. What are some common applications of the chi-square distribution in statistics?
3. How do you interpret a chi-square value in the context of hypothesis testing?
4. How can you calculate the critical value of a chi-square distribution for a given significance level?
5. What is the relationship between the chi-square and normal distributions?

**Tip:** When working with chi-square distributions, always pay attention to the degrees of freedom, as they directly affect the shape and values of the distribution.

Find the Probability P(χ²_78 > 47.79). (Round to four decimal places as needed.)

Learn how to calculate the probability P(χ²_78 > 47.79) using the chi-square distribution. This problem involves understanding probability distributions, cumulative distribution functions, and rounding to four decimal places. Suitable for undergraduate statistics students.

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