T Test - One or Two Tailed?

[karunk]

David,

I am having difficulties understanding when to use one tailed t test, and when to use two tailed. For Eg: In a practice question, 2009-Gujarati-5-8.pdf, question 7.07c, Ŷt = 5807 + 3.24Xt r² = 0.22 se = (1.634),

Compute the t value under H0:B2 = 0. Is it statistically significant at the 5 percent level? Which t test do you use, one tailed or two-tailed, and why?

Answer: t = 3.24 / 1.634 = 1.9829. Since car sales are expected to be positively related to real disposable income, the null and alternative hypotheses should be: H0 : B2 ≤ 0 and H1 : B2 > 0. Therefore, an one-tailed t test is appropriate in this case. The 5% one-tailed t value for 14 d.f. is 1.761. Since the computed t value of 1.9829 exceeds the critical value, reject the null hypothesis (one- and two tail tests sometimes give different results).

I dont quite understand this.

Also, please help me understand why d.f is k-1 at times, at k at times. I still dont seem to understand that part clearly enough.

Regards,
Arun

 

[David Harper CFA FRM]

Hi Arun,

I didn't write that question, FWIW (I *add* to the Gujarati questions, but this is a source Gujarati).
There is an acceptable convention to signify a one-tailed null, as Gujarati does, with H0:B2 = 0.
However, it seems confusing and inferior to the more specific. In this case, Gujarati means "H0:B2<=0" and it baffles me why all authors don't add the "<" or ">" to the "=" to make explicit the one-tailed.

I appreciate the authors who signify the three possible nulls with:
H0:B2 = 0
H0:B2 <= 0, or
H0:B2 >= 0

… the question otherwise, IMO, is weak (sorry). I think it would be better to ask simply (e.g) "Are car sales positively related to income?".

In this way, we can map the three possible nulls to the three questions:
Are car sales related to income?" H0:B2 = 0 and HA:B2<>0
"Are car sales positively related to income? H0:B2 <= 0 and HA: B2>0
"Are car sales negatively related to income?" H0:B2 >= 0 and HA: B2<0

… note1: our QUESTION is the ALTERNATIVE, such that our Null is the "straw man" we hope to defeat
...note2: the null always contains the "=," is an old study tip
...note3: my advice is to assume two-tailed as default because it is more common (except for VaR, VaR is always one-tailed!) and require the question to force you into a one-tail with "greater than/less than" language

As such, in my opinion, 7.07c is weakly written and I agree with your confusion. The question literally invites a two-tailed and expects too much interpretation …. A better question would be explicit that the null includes a "<."

Re df and k: yes, that is very difficult. But df = k is frankly more theoretical and you are unlikely to encounter; e.g., if we are given (k) random normals, the sum of their squares has df = k b/c we are not bootstrapping anything.

In practice, and it all cases relevant to us, you can pretty much rely on:
d.f. = k = n - # of variables

So in the case a test of a sample mean (t) or sample variance (chi), df = n - 1; i.e., univariate
In the case of a regression and the test of (partial) slope coefficient:
Two-variable (Y = X): df = n - 2
Three-variable (Y = X1 + X2): df = n-3

So, while it ignores some technical details (e.g., in a two-variable regression, df = n-2 is not due to the variables but rather fact that we have two ESTIMATES: slope and intercept, but no worries as # of estimates = # of variables), frankly, this is what I use to remember: df = n - # of variables, noting that test of sample mean and sample variance are univariate. Hope that helps

David

 

[karunk]

Thanks David for the elaborate explanation.

One more question. I have gone through your videos for level 1, and I have browsed through most of the practice spreadsheet. Something like a first revision to get a hang of complexity of the exam. I have gone through the practice questions for Foundation and I am doing the quantitative now. FYI I am doing L1 in Nov 2010.

I am not sure if that is enough, I still think I can do with a lot more practice. I am from an IT background and joined Risk Management IT recently. So, it takes time to get used to the financial jargons, and fit the theory into the practice questions. Is there anything else you would recommend practising, or just the ones you have annotated in BT would do?

 

[David Harper CFA FRM]

Hi Arun,

I am surprised you have gotten through all our questions ... most do not get that far, definitely not with weeks to spare. It sounds like you have well-prepared. IMO, you can never answer too many practice questions. I will keep adding daily (right up to the exam), but maybe more practice questions (?). If you do find additional good source, please share back, i would be interested to know, thanks.

David

 

[karunk]

Thanks David. I would then do a lot more practice of the same concepts. Being a novice to Risk Management, I think more reading will only help. Thanks for your inputs!!

At the moment BT is the best source I know.