Practice: Hypothesis Testing

Quantitative Methods. 14 question(s) in this unit's pool (2 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 MethodsHypothesis Testing
Your state on this unit Not started

Read the lesson for this unit · Back to your map

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

An analyst wants to test whether the mean daily return of a portfolio is different from zero. She formulates H0: μ = 0 versus Ha: μ ≠ 0. With a sample of 36 daily returns, a sample mean of 0.15%, and a sample standard deviation of 0.45%, the test statistic is closest to:

How sure are you?

Correct: A. The correct answer is 2.00.
B. Choosing 0.33 might tempt you if you mistakenly divide the sample mean by the sample standard deviation instead of the standard error, ignoring the sample size effect, which is crucial for calculating the test statistic correctly.
C. Choosing C because you might think the sign of the test statistic matters for the calculation itself is a trap; remember, the test statistic calculation is absolute, focusing on the magnitude of deviation from the mean, not the direction.

Unit: hypothesis-testing

Question 2Exam level

Using the data from the previous question (t-stat = 2.00, n=36, two-tailed test), at a 5% significance level, the analyst should most likely:

How sure are you?

Correct: B. The correct answer is Fail to reject H0 because 2.00 < 2.030.
A. You might use wrong df or wrong table and gets a lower critical value. The correct critical value for 35 df at 5% two-tailed is 2.030, not below 2.00.
C. You might use the z critical value (1.96) instead of the t critical value. Since σ is unknown and n=36, we use the t-distribution. Σ is unknown (we have sample standard deviation s), so we must use the t-distribution with n-1=35 df. The z-distribution critical value of 1.96 is not appropriate here.

Unit: hypothesis-testing

Question 3Exam level

A hypothesis test has a significance level of 5%. The probability of a Type II error is 20%. The power of the test is closest to:

How sure are you?

Correct: B. The correct answer is 80%.
A. Choosing 5% might tempt you if you confuse the significance level with the power of the test, but remember the power of the test is actually the probability of correctly rejecting a false null hypothesis, which here is 1 minus the probability of a Type II error, or 80%.
C. Choosing 75% might tempt you if you mistakenly subtract the significance level from the power, but remember, the power of the test is actually calculated as 1 minus the probability of a Type II error, making 80% the accurate value.

Unit: hypothesis-testing

Question 4Exam level

A researcher reports a p-value of 0.03 for a hypothesis test conducted at the 5% significance level. Which of the following conclusions is most appropriate?

How sure are you?

Correct: B. The correct answer is Reject the null hypothesis.
A. The most common p-value misconception. Interpreting p-value as the probability H0 is true. The p-value is the probability of observing a test statistic as extreme as calculated, ASSUMING H0 is true. It is NOT the probability that H0 is true. This distinction is directly tested.
C. You might see a small number and confuses direction. Small p-value means strong evidence against H0. A small p-value (below α) is evidence AGAINST H0, not in favor of it. If p < α, reject H0.

Unit: hypothesis-testing

Question 5Exam level

Which of the following best describes a Type I error in hypothesis testing?

How sure are you?

Correct: A. The correct answer is Rejecting the null hypothesis when it is actually true.
B. You might mix up Type I and Type II. Type II is failing to reject a false H0. Option B describes Type II error (β), not Type I. Type I = false positive (rejecting true null). Type II = false negative (failing to reject false null).
C. Conceptual confusion. Setting low α reduces Type I error but increases Type II. Setting α low reduces Type I error probability; it is not a definition of Type I error itself.

Unit: hypothesis-testing

Question 6Exam level

An analyst tests whether the variance of annual returns for a mutual fund equals 0.04 (σ² = 0.04). Using a sample of 25 annual returns with sample variance of 0.06, the chi-square test statistic is closest to:

How sure are you?

Correct: A. The correct answer is 36.00.
B. Choosing 24.00 might tempt you if you incorrectly used the sample mean instead of the sample variance in your calculation, but the chi-square test statistic requires the use of sample variance, leading to the correct calculation that results in 36.00.
C. Choosing 15.00 might tempt you if you incorrectly used the sample mean instead of the sample variance in your chi-square test statistic calculation, but the chi-square statistic requires using the sample variance, leading to the correct calculation of 36.00.

Unit: hypothesis-testing

Question 7Exam level

A portfolio manager tests whether the mean return of Portfolio A is greater than zero, using a one-tailed test at the 1% significance level with 30 observations and unknown population variance. The critical t-value is closest to:

How sure are you?

Correct: B. The correct answer is 2.462.
A. Choosing 2.326 might tempt you if you mistakenly used the z-table instead of the t-table, as 2.326 is the critical value for a one-tailed z-test at the 1% significance level, but for small sample sizes like 30 with unknown population variance, you must use the t-distribution, which gives a higher critical value of 2.462.
C. Choosing 2.756 might tempt you if you mistakenly used the critical value for a two-tailed test instead of a one-tailed test at the 1% significance level, which would lead to selecting a higher critical t-value than necessary for this scenario.

Unit: hypothesis-testing

Question 8Exam level

Two independent samples are drawn from populations. Sample 1 has variance s₁² = 0.09 (n₁=21) and Sample 2 has variance s₂² = 0.04 (n₂=16). An F-test for equality of variances (H0: σ₁² = σ₂²) at 5% significance (two-tailed). The F-statistic is closest to:

How sure are you?

Correct: A. The correct answer is 2.25.
B. Choosing 0.44 might seem correct if you mistakenly calculated the F-statistic by dividing the smaller variance by the larger variance, but the F-statistic requires dividing the larger variance by the smaller variance, making 0.44 incorrect.
C. Choosing 1.50 might be tempting if you mistakenly averaged the two sample variances, but the F-statistic requires you to divide the larger variance by the smaller variance, making 1.50 incorrect in this context.

Unit: hypothesis-testing

Question 9Exam level

Which statement about the relationship between Type I and Type II errors is most accurate?

How sure are you?

Correct: A. The correct answer is Decreasing the significance level increases the probability of a Type II error.
B. You might think 'lower significance = more careful = fewer mistakes of all kinds'. Lowering α reduces Type I error but INCREASES Type II error. You cannot reduce both simultaneously without increasing sample size.
C. Confusion about direction: larger n actually reduces Type II error (increases power). Increasing sample size reduces Type II error (increases power = 1-β). It does NOT increase Type II error.

Unit: hypothesis-testing

Question 10Exam level

An analyst tests whether the population mean return equals 8% (H0: μ = 8%). The sample mean is 10%, sample std dev is 6%, and n=25. The population standard deviation is unknown. The analyst should use which test, and the test statistic is closest to:

How sure are you?

Correct: A. The correct answer is t-test; t = 1.67.
B. You might see n=25 which is not > 30, but applies z-test anyway; or confuses z and t. When σ is UNKNOWN (sample std dev given), use t-test regardless of sample size. n=25 makes this even clearer. Both the unknown σ rule and small-n rule point to t-test.
C. You might be tempted to choose t = 0.33 if you incorrectly calculate the test statistic by dividing the sample mean by the sample standard deviation instead of using the correct formula which subtracts the hypothesized mean from the sample mean and divides by the standard error, leading to the correct t-value of 1.67.

Unit: hypothesis-testing

Question 11Exam level

A one-tailed test is conducted at the 5% significance level. The null hypothesis H0: μ ≤ 10 and alternative Ha: μ > 10. If the calculated test statistic is 1.72 and the critical value is 1.699, the correct conclusion is most likely:

How sure are you?

Correct: B. The correct answer is Reject H0; there is sufficient evidence that μ > 10.
A. You might be uncomfortable rejecting when the margin is small; tries to apply judgment. Hypothesis testing is binary at the stated significance level. If t-stat > critical value, reject H0. There is no 'close call' adjustment.
C. You might use incorrect CFA language. 'accept H0'. CFA curriculum NEVER uses 'accept the null hypothesis.' The correct language is always 'fail to reject H0.' Option C is wrong on both logic and language.

Unit: hypothesis-testing

Question 12Exam level

Which test statistic is most appropriate to test whether the variance of a normally distributed population equals a specified value?

How sure are you?

Correct: C. The correct answer is Chi-square statistic.
A. You might be tempted to choose the t-statistic because it is commonly used for hypothesis testing involving means, but the t-statistic is designed to test hypotheses about the mean of a population, not its variance, unlike the chi-square statistic which is specifically used for variance testing.
B. You might be tempted to choose the F-statistic because it is used for comparing variances, but the F-statistic is specifically for comparing two sample variances, not for testing a single population variance against a specified value like the chi-square statistic does.

Unit: hypothesis-testing

Question 13Above the exam

An analyst tests whether a fund's mean monthly return differs from zero at the 5% significance level and fails to reject the null hypothesis. A colleague argues this proves the fund's true mean return IS zero. Combining the logic of hypothesis testing with the concept of a Type II error, the colleague's claim is most likely:

How sure are you?

Correct: B. Hypothesis testing never proves the null hypothesis true; failing to reject it simply means the sample evidence was not strong enough, at the chosen significance level, to conclude the null is false. This is exactly the situation a Type II error describes: failing to reject a null hypothesis that is, in reality, false. The test's design cannot rule out that possibility, it can only control the probability of the OTHER kind of error (Type I, rejecting a true null) via the significance level chosen.
A. Classical hypothesis testing is built around DISPROVING (rejecting) the null hypothesis with sufficient evidence; it is never designed to statistically PROVE the null true. 'Fail to reject' and 'proven true' are different statistical claims, and confusing them is one of the most common conceptual traps on this LOS.
C. Whether the test was one-tailed or two-tailed does not change the fundamental logic that failing to reject a null hypothesis never proves it true; that limitation applies to hypothesis tests generally, not to a specific tail-count choice.

Unit: hypothesis-testing

Question 14Above the exam

A researcher runs the same hypothesis test on 20 different, unrelated fund managers' returns, each at the 5% significance level, and finds that exactly 1 manager's result is statistically significant. Applying the definition of significance level together with the logic of multiple testing, the most likely correct interpretation is that:

How sure are you?

Correct: B. A 5% significance level means that, by construction, there is a 5% chance of a Type I error (a false positive) on any single test where the null is actually true. Running 20 independent tests at 5% each, the EXPECTED number of false positives from chance alone is 20 x 0.05 = 1. Finding exactly one 'significant' result out of 20 is therefore consistent with pure random noise and provides weak, not strong, evidence of genuine skill for that one manager.
A. A single significant result out of 20 tests at the 5% level is exactly what chance alone predicts (1 expected false positive), so treating it as strong evidence of real skill ignores the multiple-testing context the question sets up.
C. There is nothing statistically invalid about applying the same significance level to several independent tests; the issue is not test validity, it is that running many tests raises the chance of at least one false positive by chance alone, which is a separate consideration from whether any individual test was performed correctly.

Unit: hypothesis-testing