Regression Analysis: Compute SSE, SST, SSR, r^2, and Correlation Coefficient

Math Problem Statement

Consider the data.

The estimated regression equation for these data is

ŷ = 68.25 − 2.75x.

(a)

Compute SSE, SST, and SSR using equations

SSE = Σ(yi − ŷi)2, SST = Σ(yi − y)2,

and

SSR = Σ(ŷi − y)2.

SSE= SST= SSR=

(b)

Compute the coefficient of determination

r2.

(Round your answer to three decimal places.)

Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.)

The least squares line provided a good fit as a large proportion of the variability in y has been explained by the least squares line.The least squares line provided a good fit as a small proportion of the variability in y has been explained by the least squares line. The least squares line did not provide a good fit as a large proportion of the variability in y has been explained by the least squares line.The least squares line did not provide a good fit as a small proportion of the variability in y has been explained by the least squares line.

(c)

Compute the sample correlation coefficient. (Round your answer to three decimal places.)

Solution

Let's go step by step to compute each of the values required for parts (a), (b), and (c).

(a) Compute SSE, SST, and SSR

Given dataLet's go step by step to compute each of the values required for parts (a), (b), and (c).

(a) Compute SSE, SST, and SSR

Given

x_i & : 3, 12, 6, 20, 14 \\ y_i & : 65, 40, 45, 15, 25 \\ \hat{y} & = 68.25 - 2.75x \\ \end{aligned}$$ We also need to calculate $$\bar{y}$$, the mean of $$y_i$$. Then we will calculate each term for SSE, SST, and SSR. #### 1. **SST (Total Sum of Squares)**: $$\text{SST} = \sum (y_i - \bar{y})^2$$ #### 2. **SSR (Regression Sum of Squares)**: $$\text{SSR} = \sum (\hat{y}_i - \bar{y})^2$$ #### 3. **SSE (Sum of Squared Errors)**: $$\text{SSE} = \sum (y_i - \hat{y}_i)^2$$ Let's begin by computing each of these. I'll calculate them now.The values for part (a) are: $$\text{SST} = 1480.0, \quad \text{SSR} = 1361.25, \quad \text{SSE} = 118.75$$ ### (b) Compute the coefficient of determination $$r^2$$ The coefficient of determination $$r^2$$ is given by: $$r^2 = \frac{SSR}{SST}$$ Let's calculate $$r^2$$.The coefficient of determination $$r^2$$ is approximately: $$r^2 = 0.920$$ ### Comment on the goodness of fit: Since $$r^2 = 0.920$$, which is greater than 0.55, the least squares line provides a good fit. This means that a large proportion of the variability in $$y$$ has been explained by the regression line. ### (c) Compute the sample correlation coefficient The sample correlation coefficient $$r$$ is the square root of $$r^2$$ (with the appropriate sign, based on the slope): $$r = \sqrt{r^2}$$ Let's compute $$r$$.The sample correlation coefficient $$r$$ is approximately: $$r = 0.959$$ This indicates a strong negative linear relationship between $$x$$ and $$y$$, consistent with the negative slope of the regression line. --- Would you like further clarification on any part of the solution, or details on how each calculation was done? Here are 5 related questions to expand your understanding: 1. How would the results change if one of the data points were significantly different? 2. What is the interpretation of the slope $$-2.75$$ in the regression equation? 3. How is $$r^2$$ related to the percentage of explained variance in the data? 4. Can you calculate confidence intervals for the regression coefficients? 5. What would be the impact on $$r^2$$ if we had more data points? **Tip**: Always check residuals to assess the fit of a regression model; large residuals might indicate an outlier or a poor fit.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Correlation

Formulas

SSE = Σ(yi − ŷi)^2
SST = Σ(yi − ȳ)^2
SSR = Σ(ŷi − ȳ)^2
r^2 = SSR / SST
r = √r^2

Theorems

Least Squares Method

Suitable Grade Level

Grades 10-12

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