TQC1 Probability and Statistics II
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Free TQC1 Probability and Statistics II Questions
1. 88. What assumption must be met when performing a chi-square test of independence?
- The sample data must be normally distributed
- Expected cell frequencies must be at least 5
- The data must be measured on an interval scale
- The samples must be dependent
Explanation
For the chi-square test to be valid, expected frequencies in each cell should generally be at least 5. This ensures that the approximation to the chi-square distribution is accurate and that results are reliable.
2. 76. What is the main difference between a one-tailed and a two-tailed test?
- A one-tailed test examines deviations in only one direction
- A two-tailed test is less powerful
- A one-tailed test cannot reject H₀
- A two-tailed test does not use a critical value
Explanation
A one-tailed test checks for an effect in only one direction (e.g., greater than or less than). A two-tailed test checks both directions (difference without specifying direction). One-tailed tests are more powerful for detecting directional effects but risk missing opposite trends.
3. 32. Which of the following best describes the union of two events, A and B?
- Outcomes that are in both A and B
- Outcomes that are in either A or B (or both)
- Outcomes that are not in A or B
- Outcomes that occur with probability zero
Explanation
The union of A and B, written as P(A or B), includes all outcomes that are part of A, B, or both. It reflects the probability that at least one of the events occurs.
4. 52. If the sample size increases while the population standard deviation remains constant, what happens to the standard error?
- It increases
- It decreases
- It stays the same
- It becomes unpredictable
Explanation
Standard error (SE = σ / √n) decreases as sample size increases. Larger samples provide more precise estimates of the population mean because they reduce variability among sample means.
5. 43. Which of the following is an example of a parameter?
- The mean of a population
- The mean of a sample
- The median of a sample
- The variance of a sample
Explanation
A parameter describes an entire population characteristic, such as the true mean or proportion. In contrast, statistics (like sample means) describe samples. Parameters are typically unknown and estimated using sample statistics.
6. 1. Which of the following best explains the difference between correlation and causation?
- Correlation shows cause and effect between two variables
- Causation can exist without any correlation
- Correlation shows an association but not a cause
- Both terms describe identical relationships
Explanation
Correlation measures how two variables move together but does not prove one causes the other. Causation requires evidence that changes in one variable directly result in changes in another. Misinterpreting correlation as causation can lead to false conclusions in research.
7. 12. A regression equation is given as ŷ = 5 + 3x. What does the number 3 represent?
- The y-intercept
- The slope, showing how y changes as x increases by 1
- The correlation coefficient
- The mean of y
Explanation
The slope (3) indicates that for every one-unit increase in x, the predicted value of y increases by 3 units. It represents the rate of change between the two variables and helps make predictions in linear regression.
8. 44. If the confidence level increases from 90% to 99%, what happens to the width of the confidence interval?
- It becomes narrower
- It becomes wider
- It remains the same
- It becomes invalid
Explanation
A higher confidence level requires capturing more of the sampling distribution, which increases the margin of error. This results in a wider confidence interval, providing greater certainty that the population parameter lies within the range.
9. 35. Which of the following statements about outliers in a regression model is true?
- Outliers always improve model accuracy
- Outliers can distort the slope and correlation coefficient
- Outliers always follow the trend of other data
- Outliers have no influence on regression analysis
Explanation
Outliers can significantly affect regression results, altering the slope and correlation coefficient. They can inflate or weaken relationships, leading to misleading interpretations if not properly identified and analyzed.
10. 51. A population has a mean of 80 and a standard deviation of 12. What is the standard error of the mean for samples of size 36?
- 1.5
- 2.0
- 2.5
- 3.0
Explanation
The standard error of the mean (SE) is calculated as σ / √n = 12 / √36 = 12 / 6 = 2. This measures how much sample means are expected to vary around the population mean when sampling repeatedly.
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