Question

Of two personnel evaluation methods, the first requires a two-hour test interview while the second can be completed in less than an hour. The scores for each of the 15 individuals who took both tests are given in the next table. $$\begin{array}{ccc}\hline \text { Applicant } & \text { Test } 1(x) & \text { Test } 2(y) \\\\\hline 1 & 75 & 38 \\\2 & 89 & 56 \\\3 & 60 & 35 \\\4 & 71 & 45 \\\5 & 92 & 59 \\\6 & 105 & 70 \\\7 & 55 & 31 \\\8 & 87 & 52 \\\9 & 73 & 48 \\\10 & 77 & 41\end{array}$$ $$\begin{array}{ccc}\hline \text { Applicant } & \text { Test } 1(x) & \text { Test } 2(y) \\\\\hline 11 & 84 & 51 \\\12 & 91 & 58 \\\13 & 75 & 45 \\\14 & 82 & 49 \\\15 & 76 & 47 \\\\\hline\end{array}$$ a. Construct a scatterplot for the data. Does the assumption of linearity appear to be reasonable? b. Find the least-squares line for the data. c. Use the regression line to predict the score on the second test for an applicant who scored 85 on Test 1 . d. Construct the ANOVA table for the linear regression relating \(y\) to \(x\).

Short answer

Answer: The main steps to analyze the relationship between two tests are: 1. Construct a scatterplot. 2. Calculate the least-squares line for the data. 3. Predict the score on Test 2 for a certain Test 1 score using the regression line. 4. Construct the ANOVA table for linear regression relating y to x.

Step-by-step solution

Construct a scatterplot

Plot the data points in a graph with Test 1 scores (x) on the x-axis and Test 2 scores (y) on the y-axis. Check if the assumption of linearity appears to be reasonable.

Calculate the least-squares line for the data

To find the least-squares line (regression line), calculate the mean of x scores and mean of y scores. Then, compute the slope (b) and the y-intercept (a) of the line using the following formulas: \(b = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}\) \(a = \bar{y} - b\cdot\bar{x}\) where \(\bar{x}\) and \(\bar{y}\) are the mean scores of the two tests, and \(x_i\) and \(y_i\) are individual scores on the tests.

Predict the score on Test 2 for a certain Test 1 score

With the regression line equation found in Step 2, predict the score on Test 2 for an applicant who scored 85 on Test 1 by plugging the value of 85 as Test 1 score into the regression line: \(y = a + b \cdot x\)

Construct the ANOVA table for linear regression relating y to x

Create an ANOVA table with the following columns: Source of Variation (Regression, Residual/Error, and Total), Sum of Squares (SS), Degrees of Freedom (df), Mean Square (MS), and F-ratio. Calculate the necessary values and perform the F-test: 1. Calculate Sum of Squares Regression (SSR): \(SSR = \sum(\hat{y}_i - \bar{y})^2\) 2. Calculate Sum of Squares Error (SSE): \(SSE = \sum(y_i - \hat{y}_i)^2\) 3. Calculate Total Sum of Squares (SST): \(SST = \sum(y_i - \bar{y})^2\) 4. Calculate Degrees of Freedom for Regression (dfR): \(dfR = 1\) 5. Calculate Degrees of Freedom for Error (dfE): \(dfE = n - 2\) 6. Calculate Degrees of Freedom for Total (dfT): \(dfT = n - 1\) 7. Calculate Mean Square Regression (MSR): \(MSR = \frac{SSR}{dfR}\) 8. Calculate Mean Square Error (MSE): \(MSE = \frac{SSE}{dfE}\) 9. Calculate F-ratio: \(F = \frac{MSR}{MSE}\) 10. Compare the calculated F-ratio to the critical F-value in the F-distribution table to test for the significance of the regression.

Key concepts

Scatterplot

A scatterplot is a type of graph used to display the relationship between two quantitative variables. It consists of points plotted on a horizontal and vertical axis, representing two sets of data. To create a scatterplot, each pair of scores is represented as a single point on the graph, with one score on the x-axis and the other score on the y-axis.

For example, in the assessment of two personnel evaluation methods, the scatterplot will help visualize any apparent relationship between scores from Test 1 and Test 2. By plotting the given data, we can check if there is a general trend or pattern, such as a linear relationship. If the points seem to form a rough line, it indicates that a linear model may be suitable to describe how Test 1 scores predict Test 2 scores. Additionally, looking at the scatterplot allows us to see if there are any outliers or unusual observations that don't fit the overall pattern.

Least-Squares Line

The least-squares line, also known as the line of best fit, is a straight line that best represents the data on a scatterplot. This line minimizes the sum of the squared differences between the observed values and the values predicted by the line. Mathematically, the line can be expressed as the equation \( y = a + b \times x \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( b \) is the slope of the line, and \( a \) is the y-intercept.

Computing the coefficients involves finding the slope \( b \) derived from the covariance of the x and y values and the variance of the x values, and then calculating the y-intercept \( a \) from the mean of the y values and the slope. This procedure optimizes the predictive power of the line, allowing for accurate predictions, such as estimating a Test 2 score for someone who scored 85 on Test 1.

ANOVA Table

The ANOVA (Analysis of Variance) table is an essential tool in regression analysis used to assess the statistical significance of the model. It decomposes the total variability of the data into components: variation due to the regression line (explained) and the residual variation (unexplained).

The table includes several important columns, such as the source of variation, sum of squares, degrees of freedom, mean squares, and the F-ratio. Each sum of squares measures a different type of variation: the regression's contribution compares the predicted scores from the overall mean, whereas the error or residual reflects the variation of the actual scores from the predicted scores. After comparing the mean squares, which normalizes the sum of squares by their degrees of freedom, the F-ratio indicates whether the regression model provides a better fit than the null model, a line with no slope.

F-test

The F-test in linear regression is used to determine whether the regression model is statistically significant. This test compares the variance explained by the model with the variance unexplained, using the values calculated in the ANOVA table.

By dividing the mean square due to the regression by the mean square due to the error, we obtain the F-ratio. If the F-ratio is significantly larger than 1, it suggests that the regression model explains a substantial part of the variation in the dependent variable. We then compare the calculated F-ratio to a critical value from the F-distribution, based on the degrees of freedom for regression and error. If the calculated F-value exceeds the critical value, we reject the null hypothesis, indicating that the model significantly predicts the dependent variable, in this case, the scores on Test 2.