Linear Prediction Rule for Exam Scores: Midterm to Final

Math Problem Statement

A professor has found that scores on the midterm exam in her classes predict scores on the final exam. The regression constant in the linear prediction rule for predicting final exam scores from midterm exam scores is

3232

and the regression coefficient is

0.50.5.

Complete parts (a) through (k) below.(a) Indicate the predictor variable.

The predictor variable is

▼ the midterm exam score.

the professor.

the final exam score.

0.5.

the professor's students.

Part 2

(b) Indicate the criterion variable.

The criterion variable is

▼ the professor's students.

the midterm exam score.

the final exam score.

the professor.

0.5.

Part 3

(c) Write the linear prediction rule for this example.

ModifyingAbove Upper Y with caretYequals=nothing plus left parenthesis nothing right parenthesis left parenthesis Upper X right parenthesisenter your response here+enter your response here(X)

(Type integers or decimals.)

Part 4

(d) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Part 5

(e) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Part 6

(f) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Part 7

(g) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Part 8

(h) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Part 9

(i) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Part 10

(j) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Part 11

(k) Figure the predicted final exam score for a student whose score on the midterm was

enter your response here

(Type an integer or a decimal.)

Solution

Part (a) Indicate the predictor variable.

The predictor variable is:

the midterm exam score.

Part (b) Indicate the criterion variable.

The criterion variable is:

the final exam score.

Part (c) Write the linear prediction rule for this example.

The linear prediction rule is:

$\hat{Y} = 32 + (0.5)X$

Part (d) Predicted final exam score for a midterm score of 20.

Using the prediction rule:

$\hat{Y} = 32 + (0.5)(20) = 32 + 10 = 42$

Predicted score: 42

Part (e) Predicted final exam score for a midterm score of 30.

$\hat{Y} = 32 + (0.5)(30) = 32 + 15 = 47$

Predicted score: 47

Part (f) Predicted final exam score for a midterm score of 40.

$\hat{Y} = 32 + (0.5)(40) = 32 + 20 = 52$

Predicted score: 52

Part (g) Predicted final exam score for a midterm score of 50.

$\hat{Y} = 32 + (0.5)(50) = 32 + 25 = 57$

Predicted score: 57

Part (h) Predicted final exam score for a midterm score of 60.

$\hat{Y} = 32 + (0.5)(60) = 32 + 30 = 62$

Predicted score: 62

Part (i) Predicted final exam score for a midterm score of 70.

$\hat{Y} = 32 + (0.5)(70) = 32 + 35 = 67$

Predicted score: 67

Part (j) Predicted final exam score for a midterm score of 90.

$\hat{Y} = 32 + (0.5)(90) = 32 + 45 = 77$

Predicted score: 77

Part (k) Predicted final exam score for a midterm score of 100.

$\hat{Y} = 32 + (0.5)(100) = 32 + 50 = 82$

Predicted score: 82

Let me know if you'd like further explanation or calculations!

5 Related Questions:

How does the regression coefficient $0.5$ affect the final prediction?
What does the regression constant $32$ represent in this context?
Can we predict final exam scores if the midterm score exceeds 100?
How would the linear prediction change if the regression coefficient increased?
What limitations exist in using this model for prediction?

Tip:

Always evaluate the range and variability of predictor and criterion variables to ensure accurate model predictions.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Predictor and Criterion Variables
Algebra

Formulas

Linear prediction rule: Ŷ = a + bX

Theorems

Regression equation theorem for prediction

Suitable Grade Level

Grades 10-12

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