CFA Level IQuantitative MethodsHard
An analyst tests H0: mu = 8% versus Ha: mu ≠ 8% for a fund's mean annual return, using a sample of n = 36 years with sample mean of 9.2% and sample standard deviation of 3.5%. At the 5% significance level, the two-tailed critical t-value is approximately ±2.030. What is the calculated t-statistic and the correct conclusion?
- At = 2.06; reject H0
- Bt = 2.06; fail to reject H0
- Ct = 0.34; fail to reject H0
- Dt = 1.20; fail to reject H0
Show answer & explanationAnswer & explanation
Correct answer: A. t = 2.06; reject H0
t = (sample mean - hypothesized mean) / (s/sqrt(n)) = (9.2 - 8) / (3.5/6) = 1.2 / 0.5833 = 2.057. Since |2.057| > 2.030 (critical value), the analyst rejects H0 at the 5% significance level.
Why the other options are wrong
- B. The t-statistic is calculated correctly, but the conclusion is wrong since 2.06 exceeds the critical value.
- C. This results from an incorrect standard error calculation (dividing by n instead of sqrt(n)).
- D. This uses the numerator (1.2) as the final t-statistic without dividing by the standard error.
t-Test for a Single Mean
The t-test for a population mean compares a sample mean to a hypothesized value, standardized by the sample standard error, to determine statistical significance.
- t = (sample mean - hypothesized mean) / (s/sqrt(n))
- Reject H0 if |t-statistic| > critical t-value
- Degrees of freedom = n - 1 for a single mean t-test
Memory trick: Divide the gap by the standard error to gauge how far you strayed.