Practice
Q1. A dataset has a mean of 50 and a standard deviation of 5. According to the empirical rule (68-95-99.7), approximately what percentage of the data falls between 45 and 55?
A. 95% B. 99.7% C. 68% D. 50%
Answer: C — The empirical rule states that for approximately normal distributions, approximately 68% of the data falls within one standard deviation of the mean. Since 45 and 55 are each one standard deviation (5) from the mean (50), approximately 68% of the data falls in this range. The range 40-60 covers two standard deviations (approximately 95%), and 35-65 covers three (approximately 99.7%).” Difficulty: hard
Q2. A fair six-sided die is rolled three times. What is the probability of getting at least one 6?
A. 1/6 B. 1/2 C. 91/216 D. 125/216
Answer: C — This is best solved using the complement rule. The probability of NOT getting a 6 on one roll is 5/6. The probability of not getting a 6 in three independent rolls is (5/6)^3 = 125/216. Therefore, the probability of getting at least one 6 is 1 - 125/216 = 91/216. The complement approach is much simpler than calculating P(exactly one 6) + P(exactly two 6s) + P(exactly three 6s).” Difficulty: hard
Q3. Two events A and B are such that P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2. Are A and B independent?
A. Yes, because P(A and B) = P(A) x P(B) B. No, because P(A and B) does not equal P(A) x P(B) C. Yes, because P(A) + P(B) = P(A or B) D. Cannot be determined from the given information
Answer: A — Two events are independent if and only if P(A and B) = P(A) x P(B). Here, P(A) x P(B) = 0.4 x 0.5 = 0.2, which equals P(A and B) = 0.2. Since the condition is satisfied, A and B are independent. Note that independence is different from mutual exclusivity — mutually exclusive events cannot be independent (unless one event has probability zero).” Difficulty: hard
Q4. A 95% confidence interval for a population mean is calculated as (12.3, 18.7). Which of the following is the correct interpretation of this interval?
A. There is a 95% probability that the population mean is between 12.3 and 18.7 B. We are 95% confident that the true population mean is between 12.3 and 18.7 C. 95% of the data falls between 12.3 and 18.7 D. If we repeated the sampling many times, 95% of the sample means would fall between 12.3 and 18.7
Answer: B — The correct interpretation is: “We are 95% confident that the true population mean is between 12.3 and 18.7.” This means that if we were to take many random samples and construct 95% confidence intervals from each, approximately 95% of those intervals would contain the true population mean. The population mean is a fixed (unknown) value, not a random variable, so we do not say there is a probability that it falls in the interval. Difficulty: hard
Q5. In a hypothesis test with H0: mu = 100 and Ha: mu > 100, a researcher obtains a p-value of 0.03. Using a significance level of alpha = 0.05, what conclusion should be drawn?
A. “Fail to reject H0; there is insufficient evidence that mu > 100 B. Reject H0; there is sufficient evidence that mu > 100 C. Accept H0; mu = 100 D. Reject H0; mu is significantly less than 100’,
Answer: B ”} correctAnswer=1 explanation=Since the p-value (0.03) is less than the significance level (alpha = 0.05), we reject the null hypothesis. This means there is sufficient evidence at the 0.05 significance level to conclude that the population mean mu is greater than 100. Note: we never 'accept' the null hypothesis; we either reject it or fail to reject it. The conclusion must be stated in context, referring to the alternative hypothesis direction." difficulty=“hard” />
Q6. A researcher wants to reduce the margin of error in a confidence interval for a population proportion. Which change would accomplish this?
A. Decreasing the confidence level from 95% to 90% B. Increasing the sample size C. Both A and B would reduce the margin of error D. Neither A nor B would reduce the margin of error
Answer: C — Both decreasing the confidence level and increasing the sample size reduce the margin of error. The margin of error for a proportion is z*(sqrt(p-hat(1-p-hat)/n)), where z* is the critical value. Decreasing the confidence level reduces z* (e.g., from 1.96 to 1.645), and increasing n reduces the fraction under the square root. Both changes produce a narrower interval. The trade-off is that a lower confidence level means less certainty that the interval captures the true parameter.” Difficulty: hard
Q7. A chi-square goodness-of-fit test is conducted with 5 categories. The calculated chi-square test statistic is 9.488. What is the approximate p-value?
A. p-value > 0.10 B. 0.05 < p-value < 0.10 C. p-value < 0.01 D. p-value = 0.05 exactly
Answer: B — For a chi-square goodness-of-fit test with k = 5 categories, degrees of freedom = k - 1 = 4. From the chi-square distribution table: the critical value for df = 4 at alpha = 0.05 is 9.488, and at alpha = 0.10 is 7.779. Since the test statistic (9.488) equals the critical value at 0.05 and exceeds the critical value at 0.10, the p-value is approximately 0.05. More precisely, p is between 0.05 and 0.10 (the test statistic falls between the critical values for these two significance levels).” Difficulty: hard
Q8. A Type I error occurs when a researcher:
A. Fails to reject a false null hypothesis B. Rejects a true null hypothesis C. Fails to reject a true null hypothesis D. Rejects a false null hypothesis
Answer: B — A Type I error (false positive) occurs when the null hypothesis is true but is rejected by the test. The probability of a Type I error is equal to the significance level alpha. A Type II error (false negative) occurs when the null hypothesis is false but the test fails to reject it. The power of a test is 1 - P(Type II error), representing the probability of correctly rejecting a false null hypothesis.” Difficulty: hard
Q9. A least-squares regression line has the equation y-hat = 3.2 + 1.5x. If x increases by 2 units, by how much does the predicted value of y change?
A. Increase by 3.0 units B. Increase by 1.5 units C. Increase by 6.2 units D. Cannot be determined without the data
Answer: A — The slope of the regression line (1.5) represents the change in the predicted value of y for each one-unit increase in x. If x increases by 2 units, the predicted y increases by 2 x 1.5 = 3.0 units. The y-intercept (3.2) is the predicted y when x = 0 and does not affect how y changes with x.” Difficulty: medium
Q10. In a regression analysis, the coefficient of determination r-squared = 0.81. Which of the following is the correct interpretation?
A. The correlation between x and y is 0.81 B. 81% of the data points fall on the regression line C. 81% of the variation in y is accounted for by the linear relationship with x D. The slope of the regression line is 0.81
Answer: C — The coefficient of determination (r-squared) represents the proportion of the total variation in the response variable y that is explained by the linear relationship with the explanatory variable x. An r-squared of 0.81 means that 81% of the variation in y is accounted for by its linear relationship with x, and the remaining 19% is due to other factors or random variation. The correlation r would be sqrt(0.81) = 0.9 (or -0.9), not 0.81.” Difficulty: hard