BSIE Probability & Statistics 3 โ Questions and Answers
Question 1: In hypothesis testing, a Type II error occurs when you:
- Reject a true null hypothesis
- Fail to reject a false null hypothesis (Correct answer)
- Accept a true null hypothesis
- Reject a false null hypothesis
Correct answer: Fail to reject a false null hypothesis
A Type II error (ฮฒ) is failing to reject Hโ when Hโ is actually false โ missing a real effect.
Question 2: A 95% confidence interval for a population mean is calculated as (47.2, 52.8). What is the sample mean?
- 47.2
- 50.0 (Correct answer)
- 52.8
- Cannot be determined
Correct answer: 50.0
The sample mean is the midpoint of the confidence interval: (47.2 + 52.8) / 2 = 50.0.
Question 3: Which probability rule states P(A โฉ B) = P(A) ร P(B) when A and B are independent?
- Addition rule
- Complement rule
- Multiplication rule (Correct answer)
- Bayes' theorem
Correct answer: Multiplication rule
The multiplication rule for independent events states that the joint probability equals the product of the individual probabilities.
Question 4: For a right-skewed distribution, which inequality is typically true?
- Mean < Median < Mode
- Mean > Median > Mode (Correct answer)
- Mean = Median = Mode
- Mode > Mean > Median
Correct answer: Mean > Median > Mode
In a right-skewed distribution, the long right tail pulls the mean above the median, which is above the mode.
Question 5: A random variable X has E[X] = 10 and Var(X) = 4. What is E[3X + 5]?
- 35 (Correct answer)
- 38
- 40
- 45
Correct answer: 35
E[3X + 5] = 3ยทE[X] + 5 = 3(10) + 5 = 35.
Question 6: In statistical process control, a control chart signals a process is out of control when:
- A point falls within the control limits
- A point falls beyond 3 standard deviations from the centerline (Correct answer)
- The process mean equals the specification limit
- Two consecutive points are on the same side
Correct answer: A point falls beyond 3 standard deviations from the centerline
Points beyond ยฑ3ฯ (3 standard deviations) from the centerline are the standard signal of an out-of-control process.
Question 7: The geometric distribution models:
- Number of successes in n trials
- Number of trials until the first success (Correct answer)
- Number of events in a time interval
- Continuous waiting time between events
Correct answer: Number of trials until the first success
The geometric distribution gives the probability that the first success occurs on the k-th trial in a sequence of Bernoulli trials.
BSIE Probability & Statistics 3
In hypothesis testing, a Type II error occurs when you: