Let X1, X2, . . , X10 be a sample from a Normal distribution with mean 25 and variance 25, and let independent of the sample Y1, Y2, . . . , Y10 come from a Normal distribution with mean 0 and variance 100. Let W = K * [ ( Sigma ( from 1 until 8 ) ( ( Xi - 25 ) / 5 ) ^ 2 / ( Sigma ( from i to 4 ) ( ( Yj ) ^ 2 ) ) ], where K is a constant. For which value of K does this equation give a F (m , n) distributed random variable and What are the values of m and n ?

Let's break down the problem step by step.

### Given Information:
- \( X_1, X_2, \ldots, X_{10} \) is a sample from a Normal distribution with mean \( 25 \) and variance \( 25 \). Therefore, \( X_i \sim N(25, 25) \).
- \( Y_1, Y_2, \ldots, Y_{10} \) is a sample from a Normal distribution with mean \( 0 \) and variance \( 100 \). Therefore, \( Y_j \sim N(0, 100) \).

### Standardization:
- Standardizing \( X_i \): \(\frac{X_i - 25}{5}\) follows \( N(0, 1) \).
- So, \( \left( \frac{X_i - 25}{5} \right)^2 \) follows a chi-squared distribution with 1 degree of freedom, denoted by \( \chi^2(1) \).

### Summation of Chi-squared Variables:
- Sum of 8 such variables, \( \sum_{i=1}^8 \left( \frac{X_i - 25}{5} \right)^2 \), will follow a chi-squared distribution with 8 degrees of freedom, denoted by \( \chi^2(8) \).

### Standardizing \( Y_j \):
- For \( Y_j \sim N(0, 100) \), \(\frac{Y_j}{10} \sim N(0, 1)\).
- Therefore, \( \frac{Y_j^2}{100} \) follows \( \chi^2(1) \).
- Sum of 4 such variables, \( \sum_{j=1}^4 \frac{Y_j^2}{100} \), will follow a chi-squared distribution with 4 degrees of freedom, denoted by \( \chi^2(4) \).
- But we are interested in \( \sum_{j=1}^4 Y_j^2 \). This follows \( 100 \cdot \chi^2(4) \).

### Defining \( W \):
\[
W = K \cdot \frac{\sum_{i=1}^8 \left( \frac{X_i - 25}{5} \right)^2}{\sum_{j=1}^4 Y_j^2}
\]
- The numerator, \( \sum_{i=1}^8 \left( \frac{X_i - 25}{5} \right)^2 \), follows a chi-squared distribution with 8 degrees of freedom, i.e., \( \chi^2(8) \).
- The denominator, \( \sum_{j=1}^4 Y_j^2 \), follows a chi-squared distribution with 400 degrees of freedom (since \( \sum_{j=1}^4 \frac{Y_j^2}{100} \sim \chi^2(4) \), so \( \sum_{j=1}^4 Y_j^2 \sim 100 \times \chi^2(4) = \chi^2(4 \times 100) \)).

### F-distribution:
For \( W \) to follow an \( F(m, n) \)-distribution, we need:
\[
W = \frac{\frac{1}{m} \cdot \text{Chi-squared with } m \text{ degrees of freedom}}{\frac{1}{n} \cdot \text{Chi-squared with } n \text{ degrees of freedom}}
\]
- Here, \( m = 8 \) (numerator degrees of freedom).
- And we want the denominator to have \( n = 4 \) degrees of freedom.

So we need to modify the denominator by dividing by 100. Thus, the value of \( K \) must be:
\[
K = \frac{1}{100}
\]

### Final Answer:
- **Value of \( K \)**: \( \frac{1}{100} \)
- **Degrees of Freedom \( m \)**: \( 8 \)
- **Degrees of Freedom \( n \)**: \( 4 \)

**Tip:** Understanding the properties of chi-squared distributions and the structure of the \( F \)-distribution is crucial for solving problems involving ratio tests.

Would you like more details on any part of the solution? Here are 5 related questions:

1. How do you derive the degrees of freedom for a sum of chi-squared random variables?
2. Can we have a visual representation of an \( F(m, n) \)-distribution?
3. What are the common uses of the \( F \)-distribution in statistical tests?
4. How do you standardize a normal variable?
5. What is the relationship between the \( t \)-distribution and the \( F \)-distribution?



Let X1, X2, . . , X10 be a sample from a Normal distribution with mean 25 and variance 25, and let independent of the sample Y1, Y2, . . . , Y10 come from a Normal distribution with mean 0 and variance 100. Let W = K * [ ( Sigma ( from 1 until 8 ) ( ( Xi - 25 ) / 5 ) ^ 2 / ( Sigma ( from i to 4 ) ( ( Yj ) ^ 2 ) ], where K is a constant. For which value of K does this equation give a F (m , n) distributed random variable and What are the values of m and n?

Find the value of K that makes W an F-distributed random variable from samples of Normal distributions with known mean and variance. This problem explores probability theory, chi-squared distribution, and F-distribution properties. Suitable for college-level students.

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