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The image you provided contains two mathematical questions related to regression and statistics. I'll go through each question, outline the steps required to solve them, and provide the solutions.

### **Question 8:**
We are given the data for two discrete variables \(X\) and \(Y\) with the following summary statistics:
\[
n = 6, \sum x = 628, \sum y = 358, \sum x^2 = 698,216, \sum y^2 = 25,314
\]
\[
\text{and the table:}
\]
\[
\begin{array}{|c|c|c|}
\hline
i & x_i & y_i \\
\hline
1 & 100 & 47 \\
2 & 106 & 51 \\
3 & 104 & 47 \\
4 & 106 & 58 \\
5 & 108 & 62 \\
6 & 104 & 61 \\
\hline
\end{array}
\]

#### **Tasks:**

1. **Fit the regression equation \( Y = b_0 + b_1X \).**
2. **Calculate the means \( \bar{x} \) and \( \bar{y} \).**
3. **Calculate the standard deviations of the observations and report the result.**
4. **Using the formula, show that the best-fitting line \( Y = b_0 + b_1X \) is estimated as: \( Y = -77.6 + 1.3X \).**
5. **Determine the residuals for the fitted regression model.**

#### **Solution to Question 8:**

1. **Regression Equation:**
   The formula for the regression line is:
   \[
   Y = b_0 + b_1X
   \]
   where:
   \[
   b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
   \]
   and:
   \[
   b_0 = \bar{y} - b_1 \bar{x}
   \]

2. **Calculate Means:**
   Mean of \(X\):
   \[
   \bar{x} = \frac{\sum x_i}{n} = \frac{628}{6} \approx 104.67
   \]

   Mean of \(Y\):
   \[
   \bar{y} = \frac{\sum y_i}{n} = \frac{358}{6} \approx 59.67
   \]

3. **Standard Deviations:**
   Standard deviation of \(X\) and \(Y\) can be calculated using the formula:
   \[
   s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}, \quad s_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n - 1}}
   \]

4. **Determine Best-Fitting Line:**
   Verify that the regression equation \(Y = -77.6 + 1.3X\) is correct by substituting the values of \(b_0\) and \(b_1\).

5. **Calculate Residuals:**
   The residuals are given by:
   \[
   e_i = y_i - (\hat{y}_i) = y_i - (b_0 + b_1 x_i)
   \]

### **Question 9:**
We are given a contingency table of observed frequencies for discrete random variables \(X\) and \(Y\).

The tasks are:
1. **Determine the marginal distributions.**
2. **Calculate the expected frequencies.**
3. **Use the chi-squared test to test for independence.**

### **Solution Steps for Question 9:**

1. **Marginal Distributions:**
   Compute the row and column totals to determine the marginal distribution.

2. **Expected Frequencies:**
   Calculate the expected frequencies for each cell using the formula:
   \[
   E_{ij} = \frac{\text{Row total} \times \text{Column total}}{\text{Grand total}}
   \]

3. **Chi-Squared Test:**
   Use the formula:
   \[
   \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}
   \]
   where \(O_{ij}\) are the observed frequencies and \(E_{ij}\) are the expected frequencies.

### **Next Steps:**
Would you like detailed calculations for each step, or is there a specific part you'd like more help with?

### **Related Questions:**

1. What is the formula for calculating the slope \(b_1\) in simple linear regression?
2. How do you interpret the residuals in a regression model?
3. What is the importance of the standard deviation in statistics?
4. How is the chi-squared test used to determine the independence of two categorical variables?
5. What are the assumptions behind using a linear regression model?

**Tip:** Always ensure data is normally distributed before applying statistical tests.

Learn how to solve regression and statistics problems with detailed step-by-step solutions. Includes fitting regression equations, calculating means and standard deviations, and using the chi-squared test for independence. Suitable for advanced undergraduate students.

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