1. What is the primary characteristic of a binomial distribution?

A. It models only continuous data B. Trials are dependent and non-identical C. It has a fixed number of independent trials with the same success probability D. It describes the time between events in a Poisson process

2. For the negative binomial distribution, how is the variance expressed?

A. Variance = rq / p² B. Variance = r p / q² C. Variance = r q / p D. Variance = r / p q

3. If P(X < 650) = 0.0606 for a normal distribution, which of the following best describes the event?

A. Producing an above-average weight B. Producing a weight close to the mean C. Producing a much lower-than-average weight D. Producing exactly 650 kg

4. In a Bernoulli distribution with probability of success \( p \), the mean is:

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

5. Which property must hold for a distribution to be called binomial?

A. Unlimited number of trials B. Variable probability of success in each trial C. Exactly two mutually exclusive outcomes per trial D. Continuous range of outcomes

6. The probability mass function of the Poisson distribution is:

A. \( P(X=x) = e^{-\lambda} \frac{\lambda^x}{x!} \) B. \( P(X=x) = \binom{n}{x} p^x (1-p)^{n-x} \) C. \( P(X=x) = p (1-p)^{x-1} \) D. \( P(X=x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)

7. When is the hypergeometric distribution typically used?

A. For modeling continuous random variables B. When sampling without replacement from a finite population C. For measuring time between events D. In experiments with independent trials

8. Which of the following is NOT a property of the Poisson distribution?

A. The average number of successes (λ) is known B. Successes occur at extremely small regions with high probability C. Outcomes are classified as success or failure D. The number of events in disjoint intervals are independent

9. What is the sum of probabilities of all possible outcomes in a sample space?

A. 0 B. 0.5 C. 1 D. Cannot be determined

10. Given a normal distribution with mean 750 and standard deviation 65, find the \( Z \)-score for \( X = 650 \).

A. -1.54 B. 1.54 C. -0.46 D. 0.46

11. For a Hypergeometric distribution with parameters \( N, K, n \), the mean is given by:

A. \( \frac{nk}{N} \) B. \( \frac{n}{k} \) C. \( \frac{NK}{n} \) D. \( nkN \)

12. How do you find the probability of "at least two" successes in a binomial distribution?

A. Sum all probabilities from 2 up to n B. 1 minus the sum of probabilities for 0 and 1 success C. Sum probabilities for exactly 2 and 3 successes only D. Calculate the probability of 2 successes only

13. Which is a true statement about the shape of the normal distribution curve?

A. It is skewed to the right B. It is bimodal C. It is symmetrical and bell-shaped D. It has no peak

14. Which of the following statements about independent events is true?

A. The occurrence of one event affects the probability of the other B. If events are independent, then P(A ∩ B) = P(A) + P(B) C. The occurrence or non-occurrence of one event does not influence the other D. Independent events cannot occur simultaneously

15. Two events \( A \) and \( B \) are mutually exclusive. What is \( P(A \cap B) \)?

A. \( P(A) + P(B) \) B. \( P(A) - P(B) \) C. 0 D. 1

16. If the mean weight of cocoa produced by 20 farmers is 750 kg with a standard deviation of 65 kg, the probability that a farmer produces more than 780 kg is closest to:

A. 0.0606 B. 0.3102 C. 0.5000 D. 0.6898

17. What is the meaning of the parameter \( p \) in a Geometric distribution?

A. Probability of failure B. Probability of success C. Mean number of failures D. Variance

18. When using a standard normal table, why do we use the standardized score Z instead of the raw variable X?

A. Because Z has mean zero and standard deviation one, simplifying calculations B. Because X values are always negative C. Because the table only lists probabilities for discrete variables D. Because Z is a complex number

19. Which of the following is true about a Binomial distribution?

A. It models number of successes in \( n \) independent trials B. Probability of success changes with each trial C. It models continuous random variables D. None of the above

20. A Negative Binomial distribution models:

A. The total number of successes in \( n \) trials B. The number of failures before a fixed number of successes C. The time between Poisson events D. The sum of two geometric random variables

21. What percentage of data lies within two standard deviations from the mean in a normal distribution?

A. 68.27% B. 95.45% C. 99.73% D. 50%

22. What is the mean (expected value) of a Poisson distribution with parameter \( \lambda \)?

A. \( \lambda \) B. \( \lambda^2 \) C. \( \sqrt{\lambda} \) D. \( 1/\lambda \)

23. Which distribution models the number of failures before the first success in independent Bernoulli trials?

A. Poisson distribution B. Negative Binomial distribution C. Geometric distribution D. Hypergeometric distribution

24. Which distribution is best suited to model the number of errors in a sample of luncheon vouchers given a fixed error probability per voucher?

A. Binomial B. Poisson C. Geometric D. Normal

25. The property that the mean, median, and mode of the normal distribution coincide is because the distribution is:

A. Skewed B. Multimodal C. Symmetrical and single peaked D. Discrete

26. In a negative binomial distribution, what does the parameter r represent?

A. Number of failures before the first success B. Number of successes before the experiment stops C. Probability of success in each trial D. Total number of trials

27. If a random variable X follows a Poisson distribution with parameter λ, what is the variance of X?

A. λ² B. λ C. 1/λ D. √λ

28. For a Negative Binomial distribution with parameters \( r \) and \( p \), the mean is:

A. \( \frac{r}{p} \) B. \( \frac{rp}{1-p} \) C. \( \frac{r(1-p)}{p} \) D. \( \frac{1}{r p} \)

29. Which of the following best defines a mutually exclusive event?

A. Events that can occur simultaneously B. Events that cannot occur at the same time C. Events which always occur together D. Events influencing each other’s probability

30. Why is the normal distribution important in statistics?

A. It models rare events only B. It approximates many different natural phenomena due to the Central Limit Theorem C. It is used only for categorical data D. It is always skewed and multimodal

31. Which of the following standardization formulas converts variable \( X \) to \( Z \)?

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

32. If events \( A \) and \( B \) are independent, what is \( P(A \cap B) \)?

A. \( P(A) + P(B) \) B. \( P(A) \times P(B) \) C. \( P(A) - P(B) \) D. \( P(A)/P(B) \)

33. Approximately what percentage of the data falls within 2 standard deviations of the mean in a normal distribution?

A. 68.27% B. 95.45% C. 99.73% D. 50%

34. What is the definition of probability for an event \( A \) in a sample space \( S \)?

A. \( P(A) = \frac{n(S)}{n(A)} \) B. \( P(A) = \frac{n(A)}{n(S)} \) C. \( P(A) = n(A) - n(S) \) D. \( P(A) = n(S) + n(A) \)

35. The variance of a Geometric distribution with parameter \( p \) is:

A. \( \frac{1-p}{p^2} \) B. \( \frac{p}{1-p} \) C. \( (1-p)p \) D. \( p^2 \)

36. In the geometric distribution, what does the random variable X represent?

A. Number of successes before a fixed number of failures B. Number of trials until the first success C. Number of failures before the first success D. Total number of successes in n trials

37. The variance of a random variable \( X \) is defined as:

A. \( E(X) \) B. \( E(X^2) - [E(X)]^2 \) C. \( [E(X)]^2 - E(X^2) \) D. \( E(X^2) + [E(X)]^2 \)

38. If \( Z \) is a standard normal variable, what is \( P(Z > 1.54) \) approximately?

A. 0.9400 B. 0.0606 C. 0.5000 D. 0.3102

39. What is the mean of a binomial distribution with parameters n and p?

A. n + p B. np C. p / n D. n / p

40. What is the probability mass function of a Binomial distribution \( X \sim Bin(n, p) \)?

A. \( P(X=x) = \binom{n}{x} p^x (1-p)^{n-x} \) B. \( P(X=x) = e^{-\lambda} \frac{\lambda^x}{x!} \) C. \( P(X=x) = p (1-p)^{x-1} \) D. \( P(X=x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)

41. What does the variance measure in a probability distribution?

A. The average value of the random variable B. The spread or dispersion of the random variable around the mean C. The maximum possible value of the variable D. The minimum value of the variable

42. The variance of a Binomial distribution with parameters \( n \) and \( p \) is:

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

43. If the probability that a farmer produces less than 650 kg of cocoa is 0.0606, what is the z-score corresponding to this weight given mean 750 kg and std deviation 65 kg?

A. +1.54 B. -1.54 C. 0.46 D. -0.46

44. For a Bernoulli distribution, what is the mean μ?

A. 1 - p B. p C. p(1-p) D. sqrt(p)

45. In the normal distribution, what proportion of the data lies within 1 standard deviation from the mean?

A. 68.27% B. 95.45% C. 99.73% D. 50%

46. Which statement is true about the mean, median, and mode of a normal distribution?

A. They are all different because the curve is skewed B. The mean is always greater than the median C. Mean, median, and mode all coincide due to symmetry D. The mode is always zero

47. If \( X \) is a random variable following a Geometric distribution with parameter \( p \), then \( P(X=x) \) is given by:

A. \( p (1-p)^x \) B. \( p (1-p)^{x-1} \) C. \( (1-p)^{x-1} \) D. \( \frac{1}{p} (1-p)^x \)

48. The total area under the curve of a probability density function equals:

A. Its variance B. Its mean C. 1 D. 0

49. How do you convert a raw score X from a normal distribution to a standard normal variate Z?

A. Z = (X - mean) / variance B. Z = (X + mean) / standard deviation C. Z = (X - mean) / standard deviation D. Z = mean / (X x standard deviation)

50. In the Poisson distribution, what is the probability of exactly x occurrences in an interval?

A. (e^-λ * λ^x) / x! B. (λ * e^x) / x! C. λ^x / (e^x * x) D. e^(-x) * x! / λ^x