CFA CFA Level 1 QUANT question

# QUANT question

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• microeconomist
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On break and figured I would share my most frustrating question of the week:

You are conducting the following hypothesis test:

Ho: population mean equals 50
Ha: population mean is not 50

for a normal population with a population standard deviation of 6. The standard confidence interval for the sample mean is (51.3,54.7). You conclude:

a) the p-value for the test is >.05
b) the p-value for the test is <.05 c) the p-value for the test cannot be determined without knowing the sample size. This one had me cussin' and drawing pictures. If you have a quick and dirty method to answer it in 1 minute or less, please share. I will post my method in a day or two.

• microeconomist
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• Undecided
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It was in a pile of QUANT notes that a family member dropped off to me. She only attended seminars in San Francisco, so a provider who holds seminars there.

• 3

That’s a weird problem, thanks for sharing @microeconomist. We’re not given alpha (significance level of the test). Not sure we can assume that it’s 5% from the multiple choices…

Here’s how I’d approach it: distribution function is normal. Confidence interval is centered on 53, so that’s the sample mean; it also has a total length of 3.4 (54.7 – 51.3), which means that z(alpha/2) = 0.283 * sqrt(n). This comes from (3.4/2)/6 = 0.283.

On the other hand, we’re “two-tail” testing for z = (53 – 50) / (6 / sqrt(n)) = 0.5 * sqrt(n). From this we conclude that z > z(alpha/2), so the test conclusion is reject the null hypothesis, or in other words p < alpha. Note that if we can conclude that alpha = 5%, then the answer is obviously (b), so that's what I'd do. (As a side note, we can also calculate n=48, since z(alpha/2)=1.96). Caveat: I've used my notes, and I'm not sure this takes less than 1 min... ðŸ™‚

• microeconomist
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• Undecided
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Edulima, our only brave soul! I learned a very useful bit from your analysis Thank you!

In trying to keep my time under 1 minute (and lack of question details) I took a qualitative approach to this question. I took “standard confidence level” as meaning the interval appropriate to the subject matter of this hypothesis test. And since population mean is generally within a “standard interval,” I rejected the null. Since rejection of the null infers alpha > probability value .: answer (b).

The answer is b. But, my answer was a very shaky, hesitant, cowardly (b) which is the best I could come up with to quickly dismiss this question and move on to ones I am more comfortable with.

In all honesty, the lack of specific info and level of inference in a question like this unnerves me. Under test conditions, I would definitely skip over and save for last.

• 2

Your quick solution was based on a interesting insight, or intuition if you prefer that word. If you don’t mind me asking, where did this question come from? These are great opportunities for us to learn and/or review material, @microeconomist, I’ll always be glad to discuss!

• 0

Drawing pictures always helped me though! And I think it’s done in less than 1 min. ðŸ™‚ happy to discuss your method