Probability and Statistics practice questions

Answer: B) 1.96

• AIncorrect. Divide by \(\sqrt{n}\).
• BCorrect. \(1.96\sigma/\sqrt{n} = 1.96(10)/10\).
• CIncorrect. Divide by \(\sqrt{n}\), not n.
• DIncorrect. Use the standard error.

Answer: C) 0.941

First reduce the parallel pumps to one equivalent block: 1 − (1 − 0.90)² = 0.99. Then multiply by the series controller: 0.99 × 0.95 ≈ 0.941.

• AIncorrect. This treats all three components as being in series: 0.90 × 0.90 × 0.95 ≈ 0.770. It ignores the redundancy of the parallel pumps.
• BIncorrect. This counts only one pump, in series with the controller: 0.90 × 0.95. It gives no credit for the second pump.
• CCorrect. The parallel pair has R = 1 − (0.10)(0.10) = 0.99. In series with the controller, R = 0.99 × 0.95 ≈ 0.941.
• DIncorrect. This is the reliability of the parallel pump pair alone. It leaves out the series controller, which lowers the system reliability.

Answer: A) 2.0

• ACorrect. \(z = (x - \mu)/\sigma = 10/5\).
• BIncorrect. Divide by σ.
• CIncorrect. That's inverted.
• DIncorrect. Recheck the formula.

Answer: D) 3

The mode is the value that occurs most often. Here 3 appears twice and every other value appears once.

• AIncorrect. 7 is the largest value. It occurs only once, so it is not the most frequent.
• BIncorrect. 4.5 is the arithmetic mean, (3+3+5+7)/4 = 18/4.
• CIncorrect. 4 is the median, the average of the two middle values (3 and 5).
• DCorrect. 3 occurs twice and every other value occurs once, so 3 is the mode.

Answer: D) 0.960

• AIncorrect. Multiply the probabilities.
• BIncorrect. That's for one part.
• CIncorrect. That's roughly at least one defective.
• DCorrect. \(0.98^2\).

Answer: D) 2 hours

• AIncorrect. That isn't the calculation.
• BIncorrect. The mean is the inverse.
• CIncorrect. That isn't the calculation.
• DCorrect. The mean is 1/λ.

Answer: D) 3

• AIncorrect. That's the expected number of good parts.
• BIncorrect. That's off by a factor of 10.
• CIncorrect. That's off by a factor of 10.
• DCorrect. Binomial mean = np.

Answer: A) 5%

The coefficient of variation is the standard deviation divided by the mean: 20 MPa / 400 MPa = 0.05, or 5%. It is a unitless measure of relative spread.

• ACorrect. CV = s / x̄ = 20 / 400 = 0.05 = 5%.
• BIncorrect. This is s² / x̄² = 400 / 160,000 = 0.0025. It squares the coefficient of variation instead of using the standard deviation itself.
• CIncorrect. This divides the variance by the mean (400 / 400 = 1). The CV uses the standard deviation, not the variance.
• DIncorrect. This flips the ratio (400 / 20 = 20). The CV is the standard deviation divided by the mean.

Answer: B) $20 to predicted cost

• AIncorrect. That's the intercept.
• BCorrect. The slope is the change in y per unit x.
• CIncorrect. The slope alone applies per unit.
• DIncorrect. The slope applies per unit.

Answer: C) 0.60

Mutually exclusive events have P(A and B) = 0, so the union is the simple sum: 0.35 + 0.25 = 0.60.

• AIncorrect. This treats the events as independent: 0.35 + 0.25 − (0.35)(0.25) ≈ 0.51. Mutually exclusive events can't occur together, so P(A and B) = 0 and nothing is subtracted.
• BIncorrect. This is the product 0.35 × 0.25 ≈ 0.09, which would be P(A and B) for independent events. It isn't the union, and here the intersection is zero.
• CCorrect. For mutually exclusive events, P(A or B) = P(A) + P(B) = 0.35 + 0.25 = 0.60.
• DIncorrect. This is just the larger of the two probabilities. Because the events never overlap, the union must include both probabilities in full.