sample question from GARP

[SamuelMartin]

How do I know that I need to apply Chi-Square in the question below? I see in the explanation: "Since you are trying to test population variance" How do I know that I'm trying to test population variance from the question? Which are the clues in the questions that lead me to determinate that I have to use Chi-Square instead of T-test?

Thank you very much

Question: sample question from GARP

SunStar is a mutual fund with a stated objective of controlling volatility, as measured by the standard
deviation of monthly returns. Given the information below, you are asked to test the hypothesis that the
volatility of SunStar’s returns is equal to 5%.
Mean Monthly Return 2.5%
Monthly Standard Deviation 4.9%
Number of Observations 30

What is the correct test to be used and what is the correct conclusion at the 5% level of significance?
a. Chi-Square test; reject the hypothesis that volatility is 5%.
b. Chi-Square test; do not reject the hypothesis that volatility is 5%.
c. t-test; reject the hypothesis that volatility is 5%.
d. t-test; do not reject the hypothesis that volatility is 5%.

Answer: b

Explanation: Since you are trying to test population variance, it is appropriate to use the Chi-Square test for the
equality of two variances:
Ho : σ2 = .0025
H1 : σ2 ≠ .0025

For 29 observations, Chi square values at probability of 0.975 and .025 are 16.04707 and 45.72229. We reject the
hypothesis if computed value is <16.04707 or >45.72229.
Since the computed value is 27.84 we do not reject the hypothesis that sample standard deviation is 5%
Topic: Quantitative Analysis
Subtopic: Statistical inference and hypothesis testing
AIMS: Define and interpret the null hypothesis and the alternative hypothesis. Define, calculate and interpret
chi-squared test of significance
Reference: Damodar Gujarati, Essentials of Econometrics, 3rd Edition (McGraw‐Hill, 2006), Chapter 5

 

[David Harper CFA FRM]

Hi @SamuelMartin.

It's a good example of a question (IMO), in part b/c it would be too easy to get overly bogged down in the details. The key is the word volatility in "you are asked to test the hypothesis that the volatility of SunStar’s returns is equal to 5%." (as volatility is standard deviation is generally interchangable with variance being its square root). If you needed the t-test, it would ask you something like "you are asked to test the hypothesis that SunStar’s average (mean) return is equal to 5%." This is an old question from Gujarati and his text (IMO) is still clearest on this. He introduced four (4) sampling distributions (they have in common they all converge to the normal)

• Normal/student's t: for test of sample mean (i.e., against hypothesis about unobserved population mean)
• Chi-squared: for test of sample variance or standard deviation (i.e., against hypothesis about unobserved population variance)
• F test: for for test of comparison between to sample variances ( a tempting alternative, fortunately not given! But we are implicitly given here a population null hypothesis rather than two samples to compare)

I like the question, too, for being technically correct with "do not reject the hypothesis that volatility is 5%." It is actually okay to use shorthand and say "accept hypothesis that volatility is 5%" but that would be imprecise shorthand. (I am going +1 star your post b/c I think this is a useful share). Thanks,

 

[SamuelMartin]

David,

Thank you so much for your reply. This is a great help for me.