TQC1 Probability and Statistics II
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Free TQC1 Probability and Statistics II Questions
In hypothesis testing, the null hypothesis (H₀) usually represents:
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The claim being tested or a statement of no effect
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A conclusion that must be true
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A significant difference between groups
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An alternative theory researchers hope to prove
Explanation
The null hypothesis (H₀) is a statement of no effect, no difference, or no relationship. It acts as the default or baseline assumption that statistical tests aim to challenge using evidence from the data.
If a test statistic falls within the nonrejection region, what should the researcher conclude?
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Reject H₀
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Fail to reject H₀
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There is a significant difference
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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.
Which of the following factors affects the width of a confidence interval?
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Sample mean only
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Sample size and standard deviation
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Population size
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Hypothesis direction
Explanation
The width of a confidence interval depends on both the sample size and standard deviation. Larger samples and smaller variability produce narrower intervals, leading to more precise estimates of population parameters.
Which of the following represents the complement of event A?
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A or B
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A and B
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All outcomes not in A
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The intersection of A and B
Explanation
The complement of event A, written as A′, includes all outcomes where A does not occur. Complements help calculate probabilities for “not” events using P(A′) = 1 – P(A).
Which of the following will increase the power of a hypothesis test?
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Using a smaller sample size
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Increasing the significance level (α)
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Decreasing the difference between means
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Increasing the population variance
Explanation
Statistical power is the probability of correctly rejecting a false null hypothesis. It increases with a larger α (less stringent cutoff), a larger sample size, or smaller population variance—each improves the test’s sensitivity to detect true effects.
What is the main difference between a one-tailed and a two-tailed test?
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A one-tailed test examines deviations in only one direction
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A two-tailed test is less powerful
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A one-tailed test cannot reject H₀
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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.
If the slope of a regression line is positive, what does that indicate about the relationship between x and y?
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As x increases, y decreases
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As x increases, y increases
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As x increases, y stays constant
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There is no linear relationship
Explanation
A positive slope indicates a direct relationship between variables — as x increases, y increases. This type of relationship is common in scenarios such as study time and grades or years of experience and income.
If the F-statistic in an ANOVA test is very large, this indicates:
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High within-group variability compared to between-group variability
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Low between-group variability
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The null hypothesis is likely true
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High between-group variability relative to within-group variability
Explanation
A large F-statistic results when the between-group variance is much greater than within-group variance, suggesting that the group means differ more than random variation alone would explain. This provides evidence against the null hypothesis.
In hypothesis testing, reducing α from 0.05 to 0.01 will:
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Increase the probability of a Type I error
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Decrease the probability of a Type I error
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Increase the chance of rejecting H₀
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Have no effect on test results
Explanation
Reducing α makes the criteria for significance stricter, decreasing the probability of making a Type I error (rejecting a true null). However, it may also slightly increase the chance of a Type II error.
In a one-way ANOVA test comparing four group means, the null hypothesis states that:
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All group means are different
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At least one group mean differs from another
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All group means are equal
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The population variances are unequal
Explanation
In one-way ANOVA, the null hypothesis (H₀) assumes that all population means are equal (μ₁ = μ₂ = μ₃ = μ₄). The alternative hypothesis (H₁) claims that at least one mean differs. ANOVA tests overall mean equality before any pairwise comparisons are made.
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