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: $n = 6, \sum x = 628, \sum y = 358, \sum x^2 = 698,216, \sum y^2 = 25,314$ $\text{and the table:}$

\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.