Probability and Statistics practice questions

Answer: D) 0.7

The expected value of a discrete random variable is the sum of each outcome times its probability: 0(0.5) + 1(0.3) + 2(0.2) = 0.7.

• AIncorrect. 1.0 is the simple average of the values 0, 1, and 2. It ignores that the outcomes have different probabilities.
• BIncorrect. 0 is the mode (the most likely outcome), not the probability-weighted mean.
• CIncorrect. 2 is the largest possible value, not the expected value.
• DCorrect. E[X] = Σx·P(x) = 0(0.5) + 1(0.3) + 2(0.2) = 0 + 0.3 + 0.4 = 0.7.

Answer: B) $12

Expected value is the sum of each outcome times its probability: 0.2(100) + 0.8(−10) = $12.

• AIncorrect. This is only the expected gain, 0.2 × 100. It leaves out the expected loss of 0.8 × 10 = $8.
• BCorrect. EV = 0.2(100) + 0.8(−10) = 20 − 8 = $12.
• CIncorrect. This adds the expected loss to the expected gain (20 + 8) instead of subtracting it.
• DIncorrect. This is only the expected loss, 0.8 × (−10). It leaves out the expected gain of $20.

Answer: A) 0.18

• ACorrect. Binomial: 2(0.1)(0.9).
• BIncorrect. There are two orderings; multiply by 2.
• CIncorrect. Include the 0.9 for the good item.
• DIncorrect. That's both defective.

Answer: C) 0.30

For independent events, the joint probability is the product of the individual probabilities: 0.6 × 0.5 = 0.30.

• AIncorrect. This is the average of the two probabilities, which has no meaning for a joint probability.
• BIncorrect. This is P(A or B) = 0.6 + 0.5 − 0.30, the probability that at least one occurs.
• CCorrect. For independent events, P(A and B) = P(A)·P(B) = 0.6 × 0.5 = 0.30.
• DIncorrect. This is the probability that neither occurs: (1 − 0.6)(1 − 0.5) = 0.20.

Answer: C) 1.96

• AIncorrect. That's for 99% two-sided.
• BIncorrect. That's a one-sided 95% value or two-sided 90%.
• CCorrect. 2.5% lies in each tail.
• DIncorrect. That covers about 68%.

Answer: B) A strong negative linear relationship

• AIncorrect. That's r near zero.
• BCorrect. As one variable rises, the other falls.
• CIncorrect. The sign shows it's negative.
• DIncorrect. Correlation doesn't prove causation.

Answer: D) 0.02

• AIncorrect. That's the chance at least one fails.
• BIncorrect. That's not "both."
• CIncorrect. They multiply.
• DCorrect. Independent probabilities multiply.

Answer: B) 3

The standard error of the mean is σ/√n. Here that is 15/√25 = 3.

• AIncorrect. This is σ/n = 15/25. The standard error divides σ by the square root of n, not by n.
• BCorrect. Standard error = σ/√n = 15/√25 = 15/5 = 3.
• CIncorrect. This is σ²/n = 225/25, which is the variance of the sample mean. The standard error is the square root of that value.
• DIncorrect. This is the population standard deviation. Sample means vary less than individual observations, by a factor of √n.

Answer: A) A Type I error, which occurs with probability α

Rejecting a true null hypothesis is a Type I error (false positive), and its probability is the significance level α. A Type II error is failing to reject a false null, with probability β.

• ACorrect. The engineers rejected a null hypothesis that was true (a false positive). The probability of doing this is the significance level α.
• BIncorrect. A Type II error means failing to reject a null hypothesis that is false. Here a true null was rejected.
• CIncorrect. 1 − β is the power, the probability of correctly rejecting a false null. This null was true, so rejecting it was a mistake.
• DIncorrect. Rejecting a true null is a Type I error, not a Type II error. The probability α belongs to the Type I error.

Answer: B) 120

• AIncorrect. Use the factorial.
• BCorrect. The number of arrangements is 5!.
• CIncorrect. Use the factorial.
• DIncorrect. That's 5!/2.