Math Problem Statement
A random sample of 145 students enrolled in statistics units in recent semesters at Macquarie University has been selected to study participation in lectures, measured as the number of polling quizzes answered, and academic performance. Some of the variables recorded for each student are listed below.
Variable Description subid Subject ID liveStream either “True”, if the student mostly watched the lectures via the live-stream, or “False”, if the student mostly came to classes numQuizzes the number of polling quizzes answered unitMark the final unit mark Address the following questions, assuming as necessary that all test assumptions are verified. Part 1 A test is run to answer the research question, ’Is there any difference between students coming to classes and live-streamers in the average number of polling quizzes answered?’ Which test is correct to use, if we know the population standard deviation?
What is the null hypothesis?
The p-value of the test is 0.24. What is the decision of the test?
Solution
Let's go through the details of the problem to address the questions systematically.
Part 1
1. Which test is correct to use, if we know the population standard deviation?
Since the research question is asking whether there is a difference between two groups (students coming to classes vs. live-streamers) in terms of the average number of polling quizzes answered, and the population standard deviation is known, the appropriate test to use is:
- Two-sample Z-test for comparing means.
This test is used to determine if there is a significant difference between the means of two independent groups when the population standard deviation is known.
2. What is the null hypothesis?
The null hypothesis (H₀) for this test is that there is no difference between the two groups in terms of the average number of polling quizzes answered. Formally, it can be stated as:
Where:
- is the mean number of quizzes answered by students who mostly came to classes.
- is the mean number of quizzes answered by students who mostly watched the lectures via live-stream.
3. The p-value of the test is 0.24. What is the decision of the test?
The p-value is a measure of the evidence against the null hypothesis.
- Decision Rule: If the p-value is less than the chosen significance level (usually ), you reject the null hypothesis. If the p-value is greater than , you fail to reject the null hypothesis.
Since the p-value is 0.24, which is greater than the common significance level of 0.05, we fail to reject the null hypothesis.
Conclusion:
There is not enough evidence to conclude that there is a significant difference between the average number of polling quizzes answered by students who mostly came to classes and those who mostly watched the lectures via live-stream.
Would you like more details on any part of the solution or have any questions?
Follow-up Questions:
- What would the alternative hypothesis be in this context?
- How would the decision change if the p-value were 0.03 instead of 0.24?
- What assumptions must be met to use a two-sample Z-test?
- How does knowing the population standard deviation affect the choice of test?
- What is the importance of the significance level () in hypothesis testing?
Tip:
When conducting hypothesis tests, always check that all assumptions (e.g., normality, independence) are satisfied before interpreting results.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Two-Sample Z-Test
P-Value
Significance Level
Formulas
Z = (X̄₁ - X̄₂) / (σ / √n₁ + σ / √n₂)
P-value comparison with α
Theorems
Central Limit Theorem
Z-Test Assumptions
Suitable Grade Level
Undergraduate
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