Problem 44 - Basic Statistics for Business and Economics with Student

Question

The following regression equation was computed from a sample of 20 observations: $$\hat{Y}=15-5 X$$ SSE was found to be 100 and SS total 400 . a. Determine the standard error of estimate. b. Determine the coefficient of determination. c. Determine the coefficient of correlation. (Caution: Watch the sign!)

Step-by-Step Solution

Answer

a. 2.36, b. 0.75, c. -0.866

Step 1: Understand the Components

We are given the regression equation \(\hat{Y} = 15 - 5X\). We also have SSE (Sum of Squares for Error) equal to 100 and SS total equal to 400. Our task is to find three things: the standard error of estimate, the coefficient of determination, and the coefficient of correlation.

Step 2: Calculate the Standard Error of Estimate

The standard error of estimate ({s_e}) is calculated using the formula:\[s_e = \sqrt{\frac{SSE}{n-2}}\]where \(n\) is the number of observations. Here, \(n=20\). Substituting the given values:\[s_e = \sqrt{\frac{100}{20-2}} = \sqrt{\frac{100}{18}} \approx 2.36\]

Step 3: Calculate the Coefficient of Determination

The coefficient of determination ({R^2}) indicates how well the regression line approximates the real data points. It is calculated as:\[R^2 = 1 - \frac{SSE}{SS_{total}}\]Substituting the given values:\[R^2 = 1 - \frac{100}{400} = 1 - 0.25 = 0.75\]

Step 4: Calculate the Coefficient of Correlation

The coefficient of correlation ({r}) is the square root of the coefficient of determination. Since we have a negative slope in our regression equation (-5), \(r\) should also be negative:\[r = -\sqrt{R^2} = -\sqrt{0.75} \approx -0.866\]

Key Concepts

Understanding Standard Error of Estimate

In regression analysis, after determining the \[\hat{Y}\] or the predicted value from our regression equation, it is crucial to understand how much the observed values deviate from these predictions. This is where the "Standard Error of Estimate" comes in handy. The standard error of estimate \(s_e\) gives us a measure of this spread or dispersion. The closer our data points are to the regression line, the smaller the standard error. To compute this, you use the formula: \[s_e = \sqrt{\frac{SSE}{n-2}}\] where \(SSE\) is the Sum of Squares for Error and \(n\) is the number of observations within your dataset. In our exercise,

• \(n\) = 20 (observations)
• \(SSE\) = 100 (given)

Therefore, our calculation becomes: \[s_e = \sqrt{\frac{100}{20-2}} = \sqrt{\frac{100}{18}} \approx 2.36\] This means that, on average, our predicted values deviate from the observed values by about 2.36 units.

Exploring the Coefficient of Determination

The Coefficient of Determination, often referred to as \(R^2\), is a vital statistic in regression analysis. It quantifies the proportion of the variance for the dependent variable that's explained by the independent variable(s) in the model. Simply put, it tells us how well our regression line represents the data. The formula for \(R^2\) is: \[R^2 = 1 - \frac{SSE}{SS_{total}}\] Where:

• \(SSE\) = Sum of Squares for Error
• \(SS_{total}\) = Total Sum of Squares

In our study,

• \(SSE\) is 100
• \(SS_{total}\) is 400

Substituting these values, we find: \[R^2 = 1 - \frac{100}{400} = 1 - 0.25 = 0.75\] An \(R^2\) of 0.75 implies that 75% of the variance in the dependent variable \(Y\) can be explained by the independent variable \(X\). An excellent goodness-of-fit means our model is quite effective. Nevertheless, always evaluate \(R^2\) in context with other tests to ensure a comprehensive understanding of the data.

Understanding Coefficient of Correlation

The Coefficient of Correlation, noted as \(r\), is a statistical indicator that measures the strength and direction of a linear relationship between two variables. Unlike \(R^2\), which measures how well data fit the model, \(r\) gives a direct measure of the relationship's linearity. \(r\) is calculated as the square root of \(R^2\): \[r = \sqrt{R^2}\] But, we must consider the direction indicated by the slope of the regression equation. Here, our regression equation has a negative slope (-5), which means we have a negative correlation. Therefore: \[r = -\sqrt{0.75} \approx -0.866\] This value indicates a strong, negative linear correlation between \(X\) and \(Y\).

• A \(r\) value near -1 means a strong negative relationship, where as \(X\) increases, \(Y\) tends to decrease.
• If \(r\) were positive, it would indicate that as \(X\) increases, \(Y\) also increases.

Understanding both the strength and the direction of \(r\) can help us make informed predictions and insights from our regression model.