### Statistical Analysis Using Regression and Hypothesis Testing**Data Overview:**The table of data consists of two variables, \( x_i \) and \( y_i \):| \( x_i \) | 1 | 2 | 3 | 4 | 5 ||-----------|---|---|---|---|---|| \( y_i \) | 4 | 8 | 6 | 11 | 13 |**Analysis Steps:**1. **Compute the Mean Square Error (MSE):** - Use the formula \( s^2 = \text{MSE} = \dfrac{\text{SSE}}{n-2} \). - Answer: 3.84 (incorrect).2. **Compute the Standard Error of the Estimate:** - Use the formula \( s = \sqrt{\text{MSE}} = \sqrt{\dfrac{\text{SSE}}{n-2}} \). - Answer: 1.958 (incorrect).3. **Compute the Estimated Standard Deviation of \( b_1 \):** - Use the formula \( s_{b_1} = \dfrac{s}{\sqrt{\sum (x_i - \bar{x})^2}} \). - Answer: 0.619 (incorrect).4. **Use the t-test for Hypothesis Testing (α = 0.05):** - Hypotheses: - \( H_0: \beta_1 = 0 \) - \( H_a: \beta_1 \neq 0 \) - Test statistic = 4.038 (incorrect). - p-value = 0.0273 (incorrect). **Conclusion:** - Correct Decision: Reject \( H_0 \). The relationship between \( x \) and \( y \) is significant.5. **Use the F-test to Confirm Hypothesis at 0.05 Level of Significance:** - **ANOVA Table:** | Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F | p-value | |---------------------|----------------|--------------------|-------------|------|---------| | Regression | 62.5 | 1 | 62.5 | 16.3 | 0.027 | | Error | 11.5

“Since you have posted a question with multiple sub parts, we will provide the solution only to the first three sub parts as per our Q&A guidelines. Please repost the remaining sub parts separately.”a) Mean square error (MSE) is calculated by using the formula : MSE = SSE/(n–2) Where, SSE = Sum of square error, and n = number of observations. SSE is calculated by using the formula, SSE = Σ(yi – ŷi)2 Also, ŷi = a + bxi ...(2) Using the data provided in the question the calculation table is as follows : xi yi (xi –x̄) (yi –ȳ) (xi –x̄)2 (xi–x̄)(yi –ȳ) 12345 4861113 -2-1012 -4.4-0.4-2.42.64.6 41014 8.80.402.69.2 Σxi = 15 Σyi = 42 Σ(xi –x̄)2 = 10 Σ(xi–x̄)(yi –ȳ) = 21 x̄ = 15/5 x̄ = 3 ȳ = 42/5 ȳ = 8.5 Slope (b) of the regression equation : b = (Σ(xi – x̄)(yi – ȳ)) /Σ(xi – x̄)2 Substituting the values from the table we get, b = 21/10 b = 2.1 y-intercept (a) of the regression equation : a = ȳ – bx̄ a = 8.4 – 2.1 × 3 a = 8.4 – 6.3 a = 2.1 Regression equation : On substituting the values of a and b in (2) we get, ŷi = 2.1 + 2.1xi Using the regression equation obtained in step 3 we have, xi yi ŷi = 2.1+ 2.1xi (yi –ŷi) (yi–ŷi)2 12345 4861113 4.26.38.410.512.6 0.2–1.72.4–0.5–0.4 0.042.895.760.250.16 Σ(yi–ŷi)2= 9.1 Now, we have SSE = 9.1 Using MSE formula provided in step 2 we get, s2 = MSE = 9.1/(5–2) s2 = MSE = 9.1/3 s2 = MSE = 3.03 (ANSWER) Part (b) : The standard error of the estimate is given by, s = √(MSE) s = √3.03 s = 1.741 (ANSWER) Part (c) : Standard deviation of b1 is calculated by using the formula, Sb1= s/√(Σ(xi–x̄)2) Substituting the values from step 3 and step 5 we have, Sb1= 1.741/√(10) Sb1 = 0.551 (ANSWER)