Math Problem Statement

A random sample of 115 students enrolled in statistics units in recent semesters at Macquarie University has been selected to study use of practice quizzes and academic performance. Some of the variables recorded for each student are listed below.

VariableDescription

subid

Subject ID

intDom

domestic or international

numQuizzes

the number of practice quizzes attempted

unitMark

the final unit mark

A simple linear regression is run to answer the research question, ’What is the relation between the final unit mark and the number of practice quizzes attempted?’ The results of the analysis are reported below:

        Estimate Std. Error t value Pr(>|t|)

(Intercept) 42.3520 2.5574 16.5604 0 numQuizzes 0.5947 0.0689 8.6312 0

Address the following questions, assuming as necessary that all test assumptions are verified.

What is the null hypothesis? Choose one among the following options:

a. There is a significant relation, linear or non-linear, between the two variables

b. The slope of the regression line is either positive or negative

c. The relation between the two variables is linear

d. The slope of the regression line is significantly different from zero

e. There is no relation whatsoever, linear or non-linear, between the two variables

f. The slope of the regression line is zero

Solution

The null hypothesis in a simple linear regression typically tests whether the slope of the regression line is zero, meaning that there is no linear relationship between the independent variable (number of practice quizzes) and the dependent variable (final unit mark).

In this context, the null hypothesis would be:

f. The slope of the regression line is zero.

This would imply that the number of practice quizzes attempted has no linear effect on the final unit mark. If the null hypothesis is rejected, it suggests that the number of quizzes does have a significant effect on the final mark.

Would you like to dive into more details or have any specific questions?

Here are 5 related questions:

  1. What would the alternative hypothesis be for this test?
  2. How do we interpret the p-value of 0 for the slope coefficient?
  3. What does the t-value of 8.6312 for numQuizzes tell us about the strength of the relationship?
  4. How would the results change if the p-value for numQuizzes was larger than 0.05?
  5. What assumptions are important to check in a simple linear regression analysis?

Tip: In linear regression, the interpretation of the slope is key. It tells us the expected change in the dependent variable (e.g., final unit mark) for every one-unit change in the independent variable (e.g., number of quizzes attempted).

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Hypothesis Testing
Statistical Inference

Formulas

y = β₀ + β₁x
t-value = (Estimate - 0) / Std. Error
p-value calculation

Theorems

Linear Regression Assumptions
Hypothesis Testing: Null and Alternative Hypotheses
T-Distribution

Suitable Grade Level

Undergraduate