Problem 13 - Calculus and its applications

Question

A professor wants to predict students' final examination scores on the basis of their midterm test scores. An equation was determined on the basis of data consisting of the scores of three students who took the same course with the same instructor the previous semester (see the following table). $$ \begin{array}{|c|c|} \hline \text { MIDTERM SCORE, } x & \text { FINAL EXAM SCORE, } y \\ \hline 70 \% & 75 \% \\ 60 & 62 \\ 85 & 89 \\ \hline \end{array} $$ a) Find the regression line, \(y=m x+b\). (Hint: The \(y\) -deviations are \(70 m+b-75,60 m+b-62,\) and so on. b) The midterm score of a student was \(81 \%\). Use the regression line to predict the student's final exam score.

Step-by-Step Solution

Answer

The regression line is \( y = 2.09x - 74.77 \), and the predicted final exam score for 81% midterm is 94.52%.

Step 1: Calculate the Means

First, calculate the mean of the midterm scores and the mean of the final exam scores. Thus, for the midterm scores, \( \bar{x} = \frac{70 + 60 + 85}{3} = 71.67\%. \) Similarly, for the final exam scores, \( \bar{y} = \frac{75 + 62 + 89}{3} = 75.33\%. \).

Step 2: Calculate Covariance and Variance

Calculate the covariance: \( \text{Cov}(x, y) = \frac{(70-71.67)(75-75.33) + (60-71.67)(62-75.33) + (85-71.67)(89-75.33)}{3} = \text{Cov}(x, y) = 206.67 \). Calculate the variance: \( \text{Var}(x) = \frac{(70-71.67)^2 + (60-71.67)^2 + (85-71.67)^2}{3} = 98.89 \).

Step 3: Calculate the Regression Coefficients

The slope \( m \) can be calculated as \( m = \frac{\text{Cov}(x, y)}{\text{Var}(x)} = \frac{206.67}{98.89} \approx 2.09 \). The intercept \( b \) is \( b = \bar{y} - m \bar{x} = 75.33 - 2.09 \times 71.67 \approx -74.77 \).

Step 4: Formulate the Regression Equation

The regression equation is \( y = 2.09x - 74.77 \). This equation can now be used to make predictions about the final exam score based on midterm scores.

Step 5: Predict the Final Exam Score

Given a midterm score of \( 81 \% \), substitute \( x = 81 \) into the regression equation: \( y = 2.09 \times 81 - 74.77 = 169.29 - 74.77 = 94.52 \). Thus, the predicted final exam score is approximately \( 94.52 \% \).

Key Concepts

Regression Line

The regression line represents the mathematical relationship between two variables in a linear fashion. Specifically, it is an equation of a straight line that best fits the data points on a scatter plot. This line helps to predict one variable based on the known value of another. The equation of a regression line is typically in the form of \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the intercept.

• The slope \(m\) indicates how much \(y\) changes for a unit change in \(x\). A positive slope means that as \(x\) increases, \(y\) also increases.
• The intercept \(b\) is the value of \(y\) when \(x\) is zero. It represents the starting point of the line on the y-axis.

The purpose of the regression line is to provide the best estimate of the dependent variable (\(y\)) given the independent variable (\(x\)). In the context of the professor's prediction, it allows forecasting students' final scores based on their midterm performances.

Covariance

Covariance is a measure of how much two variables change together. If the variables tend to show similar behavior, the covariance is positive. If one tends to increase while the other decreases, the covariance becomes negative.

• A covariance of zero indicates that the variables are not related.
• Covariance is computed by summing up the product of deviations of each pair of data points and dividing by the number of observations.

In the exercise, covariance helps determine the relationship between midterm and final scores. A positive covariance suggests that higher midterm scores are associated with higher final exam scores, which is crucial for predicting outcomes.

Variance

Variance measures the spread or dispersion of a set of data points. For a single variable, it indicates how much the data varies around its mean. In statistical terms, variance is the average of the squared differences from the mean.

• It is always non-negative because the squared deviations can’t be negative.
• A higher variance implies a wider spread of data.

Variance is used in linear regression to calculate the slope of the regression line. Knowing how much midterm scores vary allows us to understand the stability of our data and its potential influence on the final exam scores.

Prediction in Statistics

Prediction in statistics involves using data to forecast or estimate unknown values of a variable. In regression analysis, the goal is often to predict the dependent variable from the independent variable using the established linear relationship.

• Prediction relies on the regression equation, which summarizes the relationship between the known variables.
• In the professor’s example, we substitute a midterm score into the equation to estimate the final exam score.

Effective prediction can guide decision-making and expectations. If a student's midterm is known, we can approximate their final performance using the regression model. The more closely the data follows a linear pattern, the more accurate our prediction will be.