TQC1 Probability and Statistics II
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Free TQC1 Probability and Statistics II Questions
1. If P(A) = 0.7 and P(B) = 0.5 for independent events, what is P(A and B)?
- 0.2
- 0.35
- 0.5
- 1.2
Explanation
For independent events: P(A and B) = P(A) × P(B) = 0.7 × 0.5 = 0.35. This joint probability reflects the chance that both events occur simultaneously when they do not influence each other.
2. 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.
3. Which of the following situations requires a paired t-test?
- Comparing the mean test scores of two different classes
- Comparing pre-test and post-test scores of the same group of students
- Comparing the average heights of males and females
- Comparing sales between two separate stores
Explanation
A paired t-test is used when the same subjects are measured twice under different conditions, such as before-and-after studies. It evaluates whether the mean difference between paired observations is statistically significant.
4. A hypothesis test is conducted with α = 0.01, and the p-value is 0.008. What is the correct conclusion?
- Fail to reject H₀
- Reject H₀
- Increase α to 0.05
- Not enough information to decide
Explanation
Since p (0.008) < α (0.01), the result is statistically significant. This provides enough evidence to reject the null hypothesis at the 1% significance level.
5. In ANOVA, which condition must be satisfied to ensure the validity of results?
- Dependent observations
- Unequal group variances
- Independent samples and normally distributed populations
- Different sample sizes in each group
Explanation
ANOVA requires independent samples, approximate normality within each group, and equal population variances. Violating these assumptions can lead to incorrect F-values and unreliable significance results.
6. A researcher claims that the mean wait time is less than 20 minutes. What is the correct setup for the hypotheses?
- H₀: μ = 20; H₁: μ
- H₀: μ > 20; H₁: μ = 20
- H₀: μ
- H₀: μ ≠ 20; H₁: μ = 20
Explanation
The null hypothesis always includes equality. Since the claim is that the mean is less than 20, H₀: μ = 20 (no change), and H₁: μ < 20 (claim). This is a left-tailed hypothesis test.
7. A researcher sets α = 0.05 for a hypothesis test. What does α represent?
- The probability of making a Type II error
- The probability of rejecting a true null hypothesis
- The probability of accepting a false null hypothesis
- The test statistic’s critical value
Explanation
The significance level (α) is the probability of committing a Type I error — rejecting a true null hypothesis. An α of 0.05 means there is a 5% risk of concluding there is an effect when none actually exists.
8. If the correlation coefficient (r) between two variables is –0.85, what does this indicate?
- A weak positive relationship
- A strong negative relationship
- No relationship
- A perfect positive relationship
Explanation
The correlation coefficient ranges from –1 to +1. Values close to –1 indicate a strong negative relationship, where one variable increases as the other decreases. A value of –0.85 shows a strong and consistent inverse linear pattern.
9. A researcher is testing whether the average daily sales differ among three stores. The ANOVA test produces an F-statistic of 7.85 with a p-value of 0.003 at α = 0.05. Which conclusion is most appropriate?
- All stores have equal mean daily sales
- There is insufficient evidence to reject the null hypothesis
- At least one store’s mean daily sales differ significantly
- The sample sizes are too small for a valid conclusion
Explanation
Since the p-value (0.003) is less than α (0.05), the null hypothesis that all store means are equal is rejected. The result indicates significant evidence that at least one store’s mean daily sales differ from the others, though it doesn’t specify which ones — further post-hoc testing is needed.
10. 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.
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