1. When is it appropriate to use a t-distribution instead of the normal distribution for inference?

A. When the sample size is large and variance is known B. When the sample size is small and population variance is unknown C. When the population is normally distributed and variance is known D. When the data are nominal

2. For a joint probability distribution of two discrete random variables \(X\) and \(Y\), the marginal probability \(P_X(x)\) is:

A. Sum of joint probabilities over all \(y\) B. Sum of probabilities of \(Y\) only C. Equal to the conditional probability \(P(X|Y)\) D. Always zero

3. The notation \( P(A \cap B) \) denotes:

A. The probability of event A or event B occurring B. The product of probabilities of A and B C. The probability of both A and B occurring simultaneously D. The difference between probabilities of A and B

4. Why is random sampling important in statistical inference?

A. To reduce variability in the population B. To ensure the sample represents the population and reduces bias C. To guarantee sample size will be large D. To avoid the use of probability theory

5. The Binomial distribution is used when:

A. The number of trials is fixed and each trial has two outcomes B. Outcomes are continuous random variables C. Trials are dependent D. The events have more than two possible outcomes

6. The sample mean \( \bar{x} \) is:

A. Always greater than the population mean B. An unbiased estimator of the population mean C. The median of the sample D. Only used for population data

7. Which of the following best describes a Type I error in hypothesis testing?

A. Failing to reject a false null hypothesis B. Rejecting a true null hypothesis C. Rejecting a false alternative hypothesis D. Accepting a true alternative hypothesis

8. The probabilities of all mutually exclusive and exhaustive outcomes in a probability distribution must:

A. Sum to more than 1 B. Be all zero C. Sum to exactly 1 D. Sum to less than 1

9. The Normal distribution is characterized by:

A. Mean only B. Mean and median only C. Mean and variance only D. Variance only

10. The correlation coefficient \( \rho \) is defined as:

A. \( \text{Cov}(X,Y) \) B. \( \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \) C. \( \sigma_X \sigma_Y \) D. \( \text{Var}(X) + \text{Var}(Y) \)

11. What is an unbiased estimator?

A. An estimator whose expectation equals the true value of the parameter B. An estimator with minimum variance C. An estimator always larger than true value D. An estimator with maximum bias

12. When performing a chi-square test for independence, what do we test?

A. Whether two categorical variables have the same mean B. Whether two categorical variables are associated or independent C. Whether the sample size is adequate D. Whether the variables are normally distributed

13. The formula for covariance between two variables \(X\) and \(Y\) is:

A. \( E[(X - \mu_X)(Y - \mu_Y)] \) B. \( E(XY) - \mu_X - \mu_Y \) C. \( (X - \mu_X)(Y - \mu_Y) \) D. \( E(X) + E(Y) \)

14. What characterizes a binomial experiment?

A. Unlimited number of trials B. Only two possible outcomes in each trial C. Continuous outcome variable D. Outcomes are dependent across trials

15. For a binomial distribution, which of the following is true about the mean and variance?

A. Mean = np and Variance = npq B. Mean = n(1-p) and Variance = npq C. Mean = p and Variance = n D. Mean = np and Variance = p(1-p)

16. In the context of confidence intervals, what does a 95% confidence level mean?

A. 95% of the data lie within the interval B. There is a 95% chance the true parameter is in the interval C. 95% of similarly constructed intervals will contain the true parameter D. The sample mean is 95% accurate

17. The null hypothesis \(H_0\) generally represents:

A. The statement to be proven true B. The accepted assumption until evidence suggests otherwise C. The alternative claim D. The sample mean

18. A random variable \(X\) is said to be continuous if:

A. It has fixed finite values only B. It can take on any value within an interval C. It only takes integer values D. It has categorical values

19. Probability density functions (pdf) satisfy which of the following properties?

A. They can be negative for some values B. The total area under the curve equals 1 C. They only assign integer probabilities D. They provide cumulative probabilities directly

20. What is the primary property of a normal distribution?

A. Skewed to the right B. Symmetrical about the mean C. Always multimodal D. Defined only for discrete data

21. If a test statistic falls in the critical region of the sampling distribution, what conclusion should be made?

A. Fail to reject the null hypothesis B. Reject the null hypothesis C. Increase the sample size D. Accept the null hypothesis without doubt

22. The expected value \(E(X)\) is:

A. The mode of the random variable B. A measure of central tendency defined as a weighted average of all possible values C. The most frequent observed value D. Always equal to zero

23. What is the primary purpose of hypothesis testing in statistics?

A. To estimate population parameters B. To collect data randomly C. To make decisions about population parameters based on sample data D. To organize data into frequency tables

24. What is the difference between variance and covariance?

A. Variance measures spread of one variable; covariance measures relationship between two variables B. Variance is always positive; covariance is always zero C. Variance applies to nominal data; covariance applies to ordinal data D. There is no difference

25. The mean of a Binomial distribution \( B(n,p) \) is:

A. \( np \) B. \( n/p \) C. \( n + p \) D. \( p/n \)

26. The variance of a Binomial distribution \( B(n,p) \) is:

A. \( np \) B. \( np(1-p) \) C. \( n^2 p \) D. \( p(1-p) \)

27. Which condition must be met to apply the Central Limit Theorem?

A. Population must be normally distributed, regardless of sample size B. Sample size must be sufficiently large for population with any distribution C. Sample size must be small for normally distributed populations D. Population variance must be zero

28. In regression analysis, what does the coefficient of determination (R²) signify?

A. The slope of the regression line B. The proportion of variation in the dependent variable explained by the independent variable C. The correlation coefficient squared D. Both B and C

29. A 95% confidence interval means:

A. 95% of the sample values fall within this interval B. There is a 95% probability the population mean lies within the interval calculated from this sample C. 95% of populations have means in this range D. The population mean is always inside the interval

30. What is the mean of a dataset?

A. The value that appears most frequently B. The average of all data values C. The middle value when data are arranged in order D. The difference between the highest and lowest values

31. Confidence intervals provide:

A. Exact values of the population parameter B. A range within which the population parameter lies with a certain probability C. The mode of a dataset D. No information about population parameters

32. P-value is defined as:

A. The probability the null hypothesis is true B. The probability of observing data at least as extreme as the sample, assuming \(H_0\) is true C. The probability of the alternative hypothesis D. The sample variance

33. The Central Limit Theorem states that:

A. The sum of independent random variables tends toward a normal distribution as the number increases B. The median is equal to the mean for large samples C. The variance of sample means decreases with sample size D. All populations are normally distributed

34. What is the formula for standardizing a normal variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\)?

A. \( Z = \frac{X - \mu}{\sigma} \) B. \( Z = \frac{X + \mu}{\sigma} \) C. \( Z = \frac{\sigma - X}{\mu} \) D. \( Z = \frac{X - \sigma}{\mu} \)

35. What is Type I error in hypothesis testing?

A. Accepting \(H_0\) when it is false B. Rejecting \(H_0\) when it is true C. Accepting \(H_1\) when it is true D. Rejecting \(H_1\) when it is false

36. What assumption is critical for the validity of linear regression analysis?

A. Non-linear relationship between variables B. Residuals that are normally distributed with constant variance C. Independent variable has zero variance D. Dependent variable is categorical

37. The mean and variance of a Poisson distribution with parameter \(\lambda\) are:

A. Mean = \(\lambda\), Variance = \(\lambda\) B. Mean = \(\lambda^2\), Variance = \(\lambda\) C. Mean = \(\sqrt{\lambda}\), Variance = \(\lambda^2\) D. Mean = \(\lambda\), Variance = \(\lambda^2\)

38. How is variance defined in statistics?

A. Average of squared deviations from the mean B. Square root of the standard deviation C. Difference between the maximum and minimum values D. Sum of all data values divided by the number of values

39. What does \( P(A|B) \) represent in conditional probability?

A. Probability of A or B B. Probability of A given B has occurred C. Probability of A and B D. Probability of B given A has occurred

40. The standard deviation is:

A. The average value in the dataset B. The variance squared C. The square root of the variance D. The median of the dataset

41. What does a p-value represent in hypothesis testing?

A. The probability that the null hypothesis is true B. The probability of obtaining a sample statistic as extreme as the observed, assuming the null is true C. The probability of accepting the alternative hypothesis D. The error rate in data collection

42. What does increasing the sample size do to the standard error of the sample mean?

A. Increases the standard error B. Decreases the standard error C. Has no effect on the standard error D. Changes mean but not the standard error

43. Which of the following best describes a continuous random variable?

A. Takes values from a countable set B. Takes numerical values in an interval C. Only takes integer values D. Takes categorical values

44. When testing for difference between two population means, which scenario justifies using a two-sample t-test assuming equal variances?

A. When both populations have unknown and equal variances B. When only one population variance is known C. When variances between populations are different D. When both population means are equal

45. Two events A and B are independent if:

A. \( P(A \cap B) = P(A) + P(B) \) B. \( P(A \cap B) = P(A) P(B) \) C. \( P(A|B) = P(B) \) D. \( P(A \cup B) = P(A) + P(B) \)

46. What is the role of the significance level (alpha) in hypothesis testing?

A. It measures the sample size B. It defines the threshold probability for rejecting the null hypothesis C. It estimates the population mean D. It is the probability of committing a Type II error

47. The Poisson distribution is appropriate when:

A. Number of successes in fixed trials is counted B. Events occur independently over a continuous interval C. Outcomes are categorical and finite D. The population is normally distributed

48. What is the interpretation when a confidence interval for the difference between two means includes zero?

A. We reject the hypothesis that the means are equal B. There is significant evidence that one mean is greater than the other C. There is no statistically significant difference between the two means D. The sample sizes are too small

49. Which of the following is true about a confidence interval for a population mean when the sample size increases?

A. The interval becomes wider B. The interval becomes narrower C. The interval shifts but stays the same width D. The interval becomes less reliable

50. Which of the following is true for covariance \( \text{Cov}(X,Y) \)?

A. Always positive B. Measures linear relationship between two variables C. Is the same as correlation D. Is unaffected by changes in units of \(X\) or \(Y\)