CFA Level I Quantitative Methods Practice Questions

Correct answer: D. 84%

Chebyshev's inequality holds for any distribution and states that at least 1 - 1/k^2 of observations fall within k standard deviations of the mean. With k = 2.5, the bound is 1 - 1/6.25 = 0.84, or 84%.

• A. This is 1 - 1/k with k = 2.5, which uses the wrong formula and omits the square.
• B. This is the Chebyshev bound for k = 2, not 2.5.
• C. This is the approximate share within two standard deviations of a normal distribution, which cannot be assumed here.
• D. Correct: 1 - 1/(2.5)^2 = 84%.

Correct answer: D. The mean will typically be higher than the median, because the extreme high values pull the mean upward

In a right-skewed distribution, extreme values in the long right tail pull the arithmetic mean upward more than they affect the median. The typical ordering is mode < median < mean for right-skewed data.

• A. Incorrect — the mode doesn't work this way; in a typical right-skewed unimodal distribution the ordering is mode < median < mean, not mode above both.
• B. Incorrect — this states the opposite relationship; in a right-skewed distribution the mean typically exceeds the median, not the reverse.
• C. Incorrect — mean and median are only exactly equal in a symmetric distribution; skew by definition creates a gap between them.
• D. Correct — in a right-skewed (positively skewed) distribution, a few extreme high values pull the mean above the median.

Correct answer: B. 3.16

Mean = 8. Squared deviations sum to 40. Sample variance = 40/(n−1) = 40/4 = 10, so sample standard deviation = √10 ≈ 3.16. Choice A mistakenly divides by n (population convention). Choice C reports the variance, not std dev.

• A. 2.83 equals the square root of 8, which results from dividing the sum of squared deviations by n (the population convention) instead of n-1, then taking the square root.
• B. Mean = 8; squared deviations sum to 40; sample variance = 40/(5-1) = 10; sample standard deviation = square root of 10, approximately 3.16.
• C. 8.00 is the population variance (40/5), not a standard deviation — the square root step was never taken.
• D. 10.00 is the correct sample variance itself (40/4), but the square root step needed to convert variance into standard deviation was omitted.

Correct answer: C. The null hypothesis is true, but the test incorrectly rejects it.

A Type I error is the rejection of a true null hypothesis — a 'false positive.' Its probability is the test's significance level, α. A Type II error, by contrast, is the failure to reject a false null hypothesis — a 'false negative,' whose probability is denoted β.

• A. Incorrect — this describes a correct decision (true null, correctly not rejected) at the 5% level, not an error.
• B. Incorrect — this describes a correct decision (false null, correctly rejected), related to the test's power, not an error.
• C. Correct — rejecting a true null hypothesis is precisely the definition of a Type I error, with probability α.
• D. Incorrect — this describes a Type II error, not a Type I error.

• Source: CFA Institute refresher reading: Hypothesis Testing