91Ó°ÊÓ

Assume that in a sociology class, the teacher gives a midterm exam and a final exam. Assume that the association between midterm and final scores is linear. Here are the summary statistics: Midterm: Mean \(=72, \quad\) Standard deviation \(=8\) Final: Mean \(=72, \quad\) Standard deviation \(=8\) Also, \(r=0.75\) and \(n=28\). a. Find and report the equation of the regression line to predict the final exam score from the midterm score. b. For a student who gets 55 on the midterm, predict the final exam score. c. Your answer to part (b) should be higher than 55 . Why? d. Consider a student who gets a 100 on the midterm. Without doing any calculations, state whether the predicted score on the final exam would be higher, lower, or the same as 100 .

Step by step solution

01

Calculate the slope (b1) of the regression line

The formula to calculate the slope (b1) is: \(b1 = r * (SD_y/SD_x)\). Here, \(r\) is the correlation coefficient, \(SD_y\) is the standard deviation of the final scores, and \(SD_x\) is the standard deviation of the mid term scores. Substituting the given values, we get \(b1 = 0.75 * (8/8) = 0.75\).

02

Calculate the intercept (b0) of the regression line

The formula to calculate the intercept (b0) is: \(b0 = mean_y - b1 * mean_x\). Here, \(mean_y\) is the mean of the final scores and \(mean_x\) is the mean of the mid term scores. Substituting the given values, we get \(b0 = 72 - 0.75 * 72 = 18\).

03

Form the regression equation

The regression equation in the form \(y = b0 + b1*x\) becomes \(Final Score = 18 + 0.75 * MidTerm Score\).

04

Predict the final exam score for a student who scored 55 in the midterm

Substitute the midterm score of 55 into the regression equation to predict the final score. So, \(Final Score = 18 + 0.75 * 55 = 59.25\).

05

Explain why the predicted score is higher than the midterm score

The predicted score is higher because the slope of the regression line, 0.75 is greater than 0. This indicates a positive relationship between midterm and final scores, meaning that an increase in the midterm score is associated with an increase in the final score.

06

Predict the score for a student who scored 100 in the midterm

Without performing any calculation, we can infer that the predicted final score would be less than 100 due to the same reasons as step 5. However, for a precise prediction, substitution into the regression equation would be necessary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient

The correlation coefficient, often symbolized as \( r \), measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1.
If \( r \) is close to 1, it indicates a strong positive association; close to -1 indicates a strong negative association, and around 0 suggests no linear correlation.
In the given exercise, a correlation coefficient of 0.75 suggests a strong positive linear relationship between midterm and final exam scores. This means that as midterm scores increase, final scores tend to increase as well, and vice versa.
Understanding \( r \) is crucial because it indicates how reliable the predictions we make using regression analysis will be.

Linear Association

A linear association between two variables means that the relationship can be represented with a straight line on a graph. This straight line, also known as a regression line, shows how change in one variable is associated with change in another.
In our example, the relationship between the midterm and final exam scores is linear. This allows us to use a simple linear equation to predict one from the other. Linear associations are easier to analyze and often provide clear insights in data that exhibit this pattern.

• The strength of the association is indicated by the correlation coefficient.
• A strong linear association means that predictions using this line are likely to be more accurate.

Prediction in Statistics

Prediction in statistics refers to using past data to make informed guesses about future events or unknown outcomes. The process often involves using a regression line to estimate a 'y' value for a given 'x'.
In the context of the exercise, we predict the final exam score based on a given midterm score.
This involves substituting the known midterm score into our regression equation, which provides the predicted final score.
Statistical prediction is valuable in many fields because it transforms data into actionable insights.
However, it is important to remember that predictions are not guarantees. They offer estimates based on patterns within the data.

Slope and Intercept

The slope and intercept are crucial components of the regression line equation, \( y = b0 + b1 \times x \).
The slope (\( b1 \)) represents the rate of change; it shows how much the dependent variable (final score, in this case) changes for each unit change in the independent variable (midterm score). In our example, the slope is 0.75, demonstrating a positive linear relationship.
The intercept (\( b0 \)) represents the value of the dependent variable when the independent variable is zero. Here, the intercept is 18, indicating what the final score would be if the midterm score were 0.

• A positive slope indicates a positive relationship.
• The intercept can provide context but is often hypothetical, as a zero value may not exist within the data range.

Exam Score Prediction

Exam score prediction involves using the regression equation to estimate future scores based on current test results.
For instance, a student scoring 55 on the midterm has a predicted final score calculated as 18 + 0.75 \( \times 55\), which equals 59.25.
This predicted score is higher than their midterm score, illustrating the positive relationship identified by the correlation coefficient and the slope.
For a student scoring 100 on the midterm, the prediction without calculation can be inferred to be under 100 due to the slope being less than 1. However, exact predictions require calculation.

• Predictions are valuable for gauging possible future outcomes, providing an edge in academic planning.
• Such predictions, while insightful, are most reliable when there's a strong correlation and a linear association, as in this exercise.