Practice: Estimation and Inference

Quantitative Methods. 13 question(s) in this unit's pool (0 above the exam). Free up to ten a day; the coach picks which ones based on what you have already answered and when each is next due.

Quantitative MethodsEstimation and Inference
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Today's practice

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. Questions you have already answered correctly and confidently stay out of the way until they are due for review again.

Question 1Exam level

A portfolio manager takes a simple random sample of 64 stocks from a population of 500 stocks. The population has a standard deviation of returns of 8%. The standard error of the sample mean is closest to:

How sure are you?

Correct: B. The correct answer is 1.000%.
A. Divides σ by n (8%/64 = 0.125%) instead of σ/√n. Confuses the formula. Standard error divides by the square root of n, not n itself.
C. Simply uses the population standard deviation without adjusting for sample size. Standard error measures variability of the sample mean, not the population. It shrinks as n increases.

Unit: estimation-and-inference

Question 2Exam level

According to the Central Limit Theorem, as sample size most likely increases, the distribution of the sample mean:

How sure are you?

Correct: A. The correct answer is Approaches a normal distribution regardless of the shape of the population distribution.
B. Students think normality requires a normal population. This is the most common wrong answer. CLT is powerful precisely because it does NOT require a normal population. Any distribution converges to normal with sufficient sample size.
C. Confuses the sampling distribution with the t-distribution used when σ is unknown. The CLT describes convergence to normal. The t-distribution is used when σ is unknown, not as a statement of CLT.

Unit: estimation-and-inference

Question 3Exam level

A 95% confidence interval for a population mean is most likely best interpreted as:

How sure are you?

Correct: B. The correct answer is If we construct 100 such intervals using the same method, we expect 95 to contain the true population mean.
A. This is the intuitive interpretation that nearly every student defaults to. And it is wrong. Once the interval is constructed with actual numbers (e.g., [4.2, 5.8]), the population mean is fixed (unknown but fixed). The probability it is in this specific interval is either 0 or 1, not 95%.
C. Confuses the confidence interval for the mean with a prediction interval for individual observations. A confidence interval estimates the location of the population mean. It does not describe the distribution of individual sample means.

Unit: estimation-and-inference

Question 4Exam level

An analyst is constructing a confidence interval for a population mean. The population variance is unknown. The sample size is 20. The analyst should most likely use:

How sure are you?

Correct: B. The correct answer is The t-distribution with 19 degrees of freedom.
A. Knows to use a distribution with degrees of freedom but incorrectly assigns df = n rather than n-1. Degrees of freedom for the t-distribution in this context is n-1, not n. With n=20, df=19.
C. Recalls the 'n≥30' rule and incorrectly applies it. Sample size of 20 seems borderline so candidates guess z. The n≥30 CLT threshold relates to normality of the sampling distribution. The z vs t choice depends on whether σ is KNOWN. If σ is unknown, use t regardless of n.

Unit: estimation-and-inference

Question 5Exam level

An analyst doubles the sample size from 100 to 400. All else equal, the width of a 95% confidence interval for the population mean will most likely:

How sure are you?

Correct: A. The correct answer is Be reduced by half.
B. The sample size quadrupled, so candidates intuit 'the interval is 1/4 as wide'. Forgetting the square root. CI width is proportional to 1/√n, not 1/n. Quadrupling n only halves the width.
C. Confusion about the direction. You might sometimes invert the relationship. More data = more precision = narrower CI. Width moves in the opposite direction of sample size.

Unit: estimation-and-inference

Question 6Exam level

A researcher builds a model using 10 years of hedge fund return data. The sample only includes funds that are currently operating. Funds that closed down during the period were excluded. This is most likely an example of:

How sure are you?

Correct: C. The correct answer is Survivorship bias.
A. Both involve a time-based sample construction issue. You might confuse the two. Look-ahead bias occurs when information not available at the time of the decision is used in the analysis. Excluding closed funds is a selection issue, not a timing issue.
B. Data-mining involves searching for patterns. You might sometimes apply this to any 'selective' data use. Data-mining bias is about over-fitting patterns to historical data through repeated testing. Survivorship is about which entities are included in the sample.

Unit: estimation-and-inference

Question 7Exam level

An analyst uses year-end accounting data to test whether companies that reported high Q3 earnings subsequently outperformed the market. This is most likely an example of:

How sure are you?

Correct: B. The correct answer is Look-ahead bias.
A. Survivorship bias is the most commonly named bias. You might default to it. No information about excluding failed companies. The bias is purely about using information that was not yet available at decision time.
C. Sounds plausible when the sample is drawn from a specific subset of companies. Sample selection bias is about HOW the sample is drawn (non-random). Look-ahead bias is about WHEN the data becomes available relative to when the decision is simulated.

Unit: estimation-and-inference

Question 8Exam level

A sample mean of 12.5 and sample standard deviation of 6.0 are calculated from a sample of n=36 observations. The population standard deviation is unknown. A 90% confidence interval for the population mean is closest to:

How sure are you?

Correct: C. The correct answer is 12.5 ± 1.699(6.0/6) using t-distribution.
A. Uses z=1.645 for 90% (correct z value) but wrong distribution. Should be t when σ unknown. SE calculation is correct. Σ is unknown, so t-distribution must be used. Even with n=36 (≥30), the rule is t when σ unknown.
B. Uses z=1.645 AND divides by n=36 instead of √n=6. Two errors at once. Both errors: wrong distribution (z instead of t) and wrong denominator (n instead of √n in standard error formula).

Unit: estimation-and-inference

Question 9Exam level

A stratified random sample most likely differs from a simple random sample in that:

How sure are you?

Correct: B. The correct answer is The population is divided into subgroups and samples are drawn from each subgroup.
A. This is the definition of simple random sampling. You might confuse the two. In simple random sampling, every member has equal probability of selection. Stratified sampling adds the constraint that the population must first be divided into meaningful subgroups.
C. Sounds like a practical research limitation. You might mistake this for convenience sampling. Convenience sampling is a type of non-probability sampling. Stratified sampling is a rigorous probability sampling method.

Unit: estimation-and-inference

Question 10Exam level

Which of the following most accurately describes data-mining bias?

How sure are you?

Correct: B. The correct answer is Repeatedly searching a dataset for patterns until a statistically significant one appears by chance.
A. Mentioning failed companies makes candidates think survivorship bias, but including failed records is the correct approach. Excluding them would be survivorship bias. Data-mining is about the testing process, not sample composition.
C. This sounds like a research design problem. You might may associate any backtest flaw with data-mining. Using information not available at decision date is look-ahead bias. Data-mining is specifically about over-testing the same dataset for patterns.

Unit: estimation-and-inference

Question 11Exam level

The minimum sample size generally required for the Central Limit Theorem to apply so that the sampling distribution of the mean is approximately normal is closest to:

How sure are you?

Correct: B. The correct answer is 30.
A. Some statistical rules of thumb use n≥10 for approximations. You might confuse rules across contexts. N=10 is insufficient for the CLT approximation in most practical cases. CFA curriculum uses n=30 as the threshold.
C. Larger seems safer. You might choose 100 thinking 'more is better for approximations'. The CFA-testable threshold is 30. While larger n gives better approximation, 30 is the defined cutoff in the curriculum.

Unit: estimation-and-inference

Question 12Exam level

A point estimate of a population parameter is most likely described as:

How sure are you?

Correct: B. The correct answer is A single sample statistic used to estimate an unknown population parameter.
A. This is the definition of a confidence interval (interval estimate). The two concepts are directly contrasted in this LOS. A range with a stated probability is an interval estimate, not a point estimate.
C. You might confuse the parameter being estimated with the estimate itself. The true population parameter (μ) is unknown. That is why we estimate it. The point estimate is the sample-based approximation, not the true value.

Unit: estimation-and-inference

Question 13Exam level

As sample size increases, holding all other factors constant, which of the following is most accurate regarding a confidence interval?

How sure are you?

Correct: B. The correct answer is The interval becomes narrower.
A. Students intuitively think 'more data = more confidence' and map this to the confidence level percentage. The confidence level (e.g., 95%) is a design choice set before the analysis, not a function of sample size. More data improves precision (narrower interval), not the stated confidence level.
C. Reverse intuition. You might confuse more data with overcorrecting the estimate. More data improves the accuracy of the point estimate (by the law of large numbers, x-bar converges to μ as n increases).

Unit: estimation-and-inference