TQC1 Probability and Statistics II
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Free TQC1 Probability and Statistics II Questions
1. If the correlation coefficient (r) = 0, what does this mean?
- There is a perfect linear relationship
- There is no linear relationship between the variables
- The variables are inversely related
- The slope of the regression line is 1
Explanation
A correlation coefficient of 0 indicates no linear relationship between the variables. However, they may still have a nonlinear association — correlation only measures linear patterns.
2. If a p-value is less than the significance level α, what decision should be made regarding the null hypothesis?
- Fail to reject H₀
- Reject H₀
- Accept H₀
- Retest the hypothesis
Explanation
When the p-value is less than α, the observed results are unlikely under the null hypothesis, providing sufficient evidence to reject H₀. This suggests that the alternative hypothesis (H₁) is supported by the data.
3. A regression output shows a slope coefficient (b₁) = 2.3 with a standard error of 0.5. What is the calculated t-statistic for testing H₀: b₁ = 0?
- 2.8
- 4.6
- 3.6
- 5.2
Explanation
t = (b₁ – 0) / SE(b₁) = 2.3 / 0.5 = 4.6. This t-statistic is compared to the critical t-value to determine if the slope differs significantly from zero. A high t-value provides strong evidence of a meaningful linear relationship.
4. Which of the following best describes the purpose of hypothesis testing?
- To describe characteristics of a sample
- To make inferences about population parameters based on sample data
- To identify all possible sample outcomes
- To calculate exact population parameters
Explanation
Hypothesis testing uses sample data to evaluate claims about population parameters. It provides a formal framework for making inferences, determining whether evidence supports rejecting or retaining a null hypothesis.
5. Which of the following best defines a Type II error?
- Rejecting a true null hypothesis
- Failing to reject a false null hypothesis
- Rejecting a false null hypothesis
- Accepting a true null hypothesis
Explanation
A Type II error occurs when a false null hypothesis is not rejected — in other words, when the test fails to detect a real effect or difference. Reducing the likelihood of Type II errors often requires increasing sample size or statistical power.
6. If P(A) = 0.3 and P(B) = 0.5, and A and B are independent, what is P(A and B)?
- 0.15
- 0.20
- 0.25
- 0.30
Explanation
For independent events, P(A and B) = P(A) × P(B) = 0.3 × 0.5 = 0.15. Independence means one event does not influence the other, so the joint probability equals the product of individual probabilities.
7. Which of the following represents mutually exclusive events?
- Drawing a red card and drawing a face card from a deck
- Rolling a 3 or an even number on a single die
- Selecting a student who is both male and left-handed
- Rolling a 2 and rolling a 5 on the same single die
Explanation
Mutually exclusive (disjoint) events cannot occur simultaneously. Rolling a 2 and a 5 on the same single die is impossible—only one number appears per roll—making these events mutually exclusive.
8. Which of the following represents a joint probability?
- P(A or B)
- P(A) + P(B)
- P(A and B)
- P(A|B)
Explanation
Joint probability expresses the likelihood that two events occur together, written as P(A and B). It combines probabilities of both outcomes happening simultaneously, essential for understanding intersections between events.
9. When interpreting the slope of a regression line, what does the slope represent?
- The predicted value when x = 0
- The change in y for every one-unit increase in x
- The correlation coefficient
- The average of all x-values
Explanation
The slope shows how much the dependent variable (y) changes for each one-unit increase in the independent variable (x). A positive slope indicates that y increases with x, while a negative slope means y decreases as x increases.
10. If a test statistic falls within the nonrejection region, what should the researcher conclude?
- Reject H₀
- Fail to reject H₀
- There is a significant difference
- The test is invalid
Explanation
If the test statistic lies within the nonrejection region, the data do not provide enough evidence to reject H₀. The observed result is consistent with what would be expected if the null hypothesis were true.
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