TQC1 Probability and Statistics II

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Your Key to Passing TQC1 Probability and Statistics II : Instant Access to Test Practice Questions

Free TQC1 Probability and Statistics II Questions

1. 29. A scatterplot shows a cluster of points that form a curved pattern. What does this suggest?
  • A strong linear relationship
  • A nonlinear relationship
  • No correlation
  • A negative linear relationship

Explanation

A curved pattern in a scatterplot suggests a nonlinear relationship — the variables are related, but not in a straight-line manner. Polynomial or exponential models might fit better than simple linear regression in such cases.
2. 66. In hypothesis testing, reducing α from 0.05 to 0.01 will:
  • Increase the probability of a Type I error
  • Decrease the probability of a Type I error
  • Increase the chance of rejecting H₀
  • 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.
3. 90. In a one-way ANOVA test comparing four group means, the null hypothesis states that:
  • All group means are different
  • At least one group mean differs from another
  • All group means are equal
  • 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.
4. 54. 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. 74. A Type I error occurs when:
  • A true null hypothesis is rejected
  • A false null hypothesis is rejected
  • A true null hypothesis is accepted
  • A false null hypothesis is not rejected

Explanation

A Type I error happens when we reject a true null hypothesis — concluding that there is an effect when none exists. It’s often denoted by α, the significance level, and represents the false-positive rate in hypothesis testing.
6. 5. Which of the following best describes Simpson’s Paradox?
  • A positive correlation always means causation
  • Trends that appear in separate groups reverse when data are combined
  • Correlation coefficients are always between 0 and 1
  • Probabilities can exceed 1 when sample size is large

Explanation

Simpson’s Paradox occurs when a trend that appears in individual groups reverses when those groups are combined. It highlights how confounding variables can distort overall conclusions, reminding researchers to analyze subgroup data carefully.
7. 56. The Central Limit Theorem is important in statistics because it:
  • Describes the shape of the population distribution
  • Allows us to use the normal distribution to approximate the sampling distribution of the mean
  • Guarantees that sample means are unbiased
  • Explains why population variances increase

Explanation

The Central Limit Theorem (CLT) states that, for sufficiently large samples, the sampling distribution of the sample mean will be approximately normal regardless of the population’s distribution. It’s the foundation of inferential statistics.
8. 71. A researcher computes a 95% confidence interval for a population mean as (45.6, 54.4). What is the correct interpretation?
  • There is a 95% probability the true mean equals 50
  • There is a 95% probability that the sample mean is within this interval
  • We are 95% confident that the population mean lies between 45.6 and 54.4
  • 95% of the data values fall between 45.6 and 54.4

Explanation

A 95% confidence interval means that if many samples were taken, approximately 95% of the resulting confidence intervals would contain the true population mean. It expresses confidence in the procedure, not in a specific sample.
9. 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.
10. 25. A researcher finds r = 0.02 between time spent watching TV and exam scores. What does this mean?
  • There is a strong positive relationship
  • There is a weak negative relationship
  • There is almost no linear relationship
  • There is a strong negative relationship

Explanation

A correlation coefficient of 0.02 is very close to zero, indicating almost no linear relationship between TV time and exam scores. The variables may be unrelated or related in a non-linear way.

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