TRB PG Assistant - Sampling distributions - t, F, χ² and their uses in significance testing, large-sample tests for means and proportions Question Bank with Options

1. What is the primary difference between the t-distribution and the standard normal (Z) distribution?

A) The t-distribution has a fixed shape, while the Z-distribution varies with sample size.

B) The Z-distribution has fatter tails than the t-distribution.

C) The t-distribution is flatter and has heavier tails than the Z-distribution, especially for small degrees of freedom.

D) The t-distribution is used for large samples, while the Z-distribution is for small samples.

2. The F-statistic is always non-negative because it is a ratio of:

A) Sample means

B) Sample variances

C) Sample proportions

D) Observed frequencies

3. Which of the following conditions is essential for using a large-sample Z-test for a proportion?

A) Population standard deviation is known.

B) Sample size is small (n < 30).

C) Both n*p and n*(1-p) are greater than or equal to 10.

D) Population is normally distributed.

4. A researcher is testing if the observed frequencies of colors in a bag of candies match the company's stated proportions. This scenario would typically use a:

A) t-test for means

B) F-test for variances

C) Chi-squared goodness-of-fit test

D) Z-test for proportions

5. The degrees of freedom for a one-sample t-test are:

A) Sample size

B) Sample size - 1

C) Population size - 1

D) Number of groups - 1

6. Which sampling distribution is used when testing the ratio of two population variances?

A) t-distribution

B) Chi-squared distribution

C) F-distribution

D) Normal distribution

7. In a two-tailed Z-test for a proportion, if the calculated p-value is 0.001 and alpha is 0.05, what is the conclusion?

A) Fail to reject H₀

B) Reject H₀

C) The test is invalid

D) Cannot conclude without the test statistic

8. The distribution of the sample variance, under the assumption of normality, follows a:

A) t-distribution

B) F-distribution

C) Chi-squared distribution

D) Normal distribution

9. Which test is used to determine if a single population variance is equal to a hypothesized value?

A) F-test

B) t-test

C) Chi-squared test

D) Z-test

10. The standard error of the mean (SEM) is calculated as:

A) Population standard deviation / sqrt(sample size)

B) Sample standard deviation / sqrt(sample size)

C) Population standard deviation * sqrt(sample size)

D) Sample standard deviation * sqrt(sample size)

11. A researcher uses a large sample (n=1000) to test a hypothesis about a population mean. The population standard deviation is unknown. What distribution should ideally be used for the test statistic?

A) Z-distribution

B) t-distribution with 999 df

C) t-distribution with 1000 df

D) F-distribution

12. Which of these tests is most sensitive to outliers?

A) Chi-squared test

B) F-test

C) t-test

D) Z-test

13. When comparing variances of two populations using the F-test, the null hypothesis is typically H₀: σ₁² = σ₂². The alternative hypothesis could be:

A) H₁: σ₁² ≠ σ₂²

B) H₁: σ₁² > σ₂²

C) H₁: σ₁² < σ₂²

D) All of the above

14. The Chi-squared statistic is always:

A) Zero or positive

B) Negative or zero

C) Positive

D) Between -1 and 1

15. In the context of hypothesis testing, the 'critical region' is:

A) The range of values where the null hypothesis is likely true

B) The range of values for which the null hypothesis is rejected

C) The range of values for which the alternative hypothesis is rejected

D) The range of values where the p-value is greater than alpha

16. Which distribution is characterized by having two parameters: the mean (μ) and the variance (σ²)?

A) t-distribution

B) F-distribution

C) Chi-squared distribution

D) Normal distribution

17. The t-statistic is calculated as the difference between the sample mean and the population mean, divided by:

A) The population standard deviation

B) The sample standard deviation

C) The standard error of the mean

D) The square root of the sample size

18. What is the shape of the F-distribution?

A) Symmetrical and bell-shaped

B) Skewed to the left

C) Skewed to the right and non-negative

D) Uniform

19. If a researcher performs a t-test and obtains a p-value of 0.03, and their significance level is 0.05, what conclusion should they draw?

A) There is not enough evidence to reject the null hypothesis.

B) There is statistically significant evidence to reject the null hypothesis.

C) The result is inconclusive.

D) A Type II error has been made.

20. A Chi-squared goodness-of-fit test is used to compare observed frequencies with:

A) Sample means

B) Expected frequencies from a hypothesized distribution

C) Population variances

D) Sample standard deviations

21. When conducting an ANOVA test with 3 groups and a total sample size of 60, what are the degrees of freedom for the within-group variance (denominator)?

A) 59

B) 57

C) 3

D) 2

22. When conducting an ANOVA test with 3 groups and a total sample size of 60, what are the degrees of freedom for the between-group variance (numerator)?

A) 59

B) 57

C) 3

D) 2

23. A marketing firm wants to know if there is a significant difference in the proportion of customers who prefer product A versus product B. They survey 500 customers. Which test would be appropriate for large samples?

A) Chi-squared test for independence

B) F-test for variances

C) Z-test for proportions

D) t-test for means

24. A researcher wants to test if a new drug significantly reduces blood pressure. They have a sample of 25 patients and the population standard deviation of blood pressure is unknown. Which test is most appropriate?

A) Z-test for means

B) Paired t-test

C) Independent samples t-test

D) One-sample t-test

25. In a large-sample test for a population mean (μ) when the population standard deviation (σ) is unknown, what is used as an estimate for σ?

A) The sample mean (x̄)

B) The sample range

C) The sample standard deviation (s)

D) A pre-determined value based on prior research

26. Which of the following is NOT a use of the Chi-squared distribution?

A) Testing the goodness-of-fit of a theoretical distribution to observed data

B) Testing for independence between two categorical variables

C) Testing the equality of variances of two populations

D) Testing hypotheses about a single population variance

27. Which of the following is NOT a use of the F-distribution?

A) Analysis of Variance (ANOVA)

B) Testing the equality of two population variances

C) Goodness-of-fit tests for categorical data

D) Regression analysis (testing significance of regression coefficients)

28. Which of the following is NOT a use of the t-distribution?

A) Confidence intervals for a population mean with unknown population standard deviation

B) Testing the difference between two independent sample means

C) Testing the equality of variances of two populations

D) Testing the equality of means of two or more populations (ANOVA)

29. The power of a statistical test is defined as the probability of:

A) Making a Type I error

B) Making a Type II error

C) Correctly rejecting a false null hypothesis

D) Correctly accepting a true null hypothesis

30. A Type II error occurs when:

A) A true null hypothesis is rejected

B) A false null hypothesis is accepted

C) A true alternative hypothesis is accepted

D) A false alternative hypothesis is rejected

31. A Type I error occurs when:

A) A true null hypothesis is rejected

B) A false null hypothesis is accepted

C) A true alternative hypothesis is rejected

D) A false alternative hypothesis is accepted

32. What is the significance level (alpha) commonly set to in many statistical tests?

A) 0.01

B) 0.05

C) 0.10

D) 0.50

33. If the p-value is less than the chosen significance level (alpha), what is the typical decision regarding the null hypothesis?

A) Fail to reject the null hypothesis

B) Reject the null hypothesis

C) Accept the null hypothesis

D) The test is inconclusive

34. The 'p-value' in hypothesis testing represents:

A) The probability of the null hypothesis being true

B) The probability of observing a test statistic as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true

C) The significance level of the test

D) The power of the test

35. What is meant by the 'null hypothesis' in significance testing?

A) A statement of no effect or no difference

B) A statement of a significant effect or difference

C) The alternative hypothesis

D) The conclusion of the test

36. For a large-sample test of a population mean, when is the t-distribution preferred over the Z-distribution?

A) When the sample size is very large (n > 30)

B) When the population standard deviation is known

C) When the population standard deviation is unknown and estimated from the sample

D) When the population is known to be normally distributed

37. A large-sample test for a population proportion typically uses which distribution under the null hypothesis?

A) t-distribution

B) F-distribution

C) Chi-squared distribution

D) Normal (Z) distribution

38. Which sampling distribution is appropriate for testing the equality of variances between two populations?

A) t-distribution

B) Z-distribution

C) F-distribution

D) Chi-squared distribution

39. What is a key assumption for using the Chi-squared distribution in tests of independence?

A) Data must be normally distributed

B) Expected frequencies in each cell should not be too small

C) Population variances must be equal

D) Sample size must be small

40. The degrees of freedom for a Chi-squared test of independence are calculated as:

A) Number of categories - 1

B) (Number of rows - 1) * (Number of columns - 1)

C) Sample size - 1

D) Number of groups - 1

41. A Chi-squared test for independence is used to determine if there is an association between:

A) Two continuous variables

B) A continuous variable and a categorical variable

C) Two categorical variables

D) Two population variances

42. The Chi-squared (χ²) distribution is primarily used for testing hypotheses about:

A) Population means

B) Population variances

C) Population proportions and independence of categorical variables

D) Differences between two population means

43. In an ANOVA test, what does a significant F-statistic suggest?

A) All group means are equal

B) At least one group mean is different from the others

C) The sample variances are equal

D) The null hypothesis is always accepted

44. The F-statistic is calculated as the ratio of:

A) Sample variance to population variance

B) Mean square error within groups to mean square error between groups

C) Mean square error between groups to mean square error within groups

D) Observed frequencies to expected frequencies

45. What is the primary use of the F-distribution in statistical testing?

A) Testing the difference between two sample means

B) Testing the variance of a single population

C) Comparing the variances of two populations

D) Testing the goodness-of-fit of a distribution

46. Which statistical test uses the t-distribution to compare the means of two independent samples?

A) Paired t-test

B) Independent samples t-test

C) ANOVA

D) Chi-squared test

47. As the degrees of freedom increase, the t-distribution approaches which other distribution?

A) Chi-squared distribution

B) F-distribution

C) Normal (Z) distribution

D) Binomial distribution

48. The t-distribution is characterized by its 'degrees of freedom'. What does the number of degrees of freedom represent in the context of the t-distribution?

A) The size of the population

B) The sample size minus one

C) The number of variables in the data

D) The level of significance chosen

49. Which sampling distribution is typically used for small sample sizes when the population standard deviation is unknown?

A) Z-distribution

B) F-distribution

C) Chi-squared distribution

D) t-distribution

50. What is the primary purpose of a sampling distribution?

A) To describe the population parameters

B) To represent the distribution of sample statistics

C) To calculate individual data point probabilities

D) To determine the sample size

Sampling distributions - t, F, χ² and their uses in significance testing, large-sample tests for means and proportions - Question Bank