Typo's P1.T2.72 - Student's t distribution - Question # 72.2, Page 105/106

[Dr. Jayanthi Sankaran]

Hi David,

Q# 72.2: Do we reject the null hypothesis that the true population average FICO score is 690 at, respectively, 95% and 99% confidence; i.e., Null Ho : population average = 690 such that Alternative H1: population average <> 690?

A 72.2. C. (Yes and no)
At 19 degrees of freedom, 95% two-tailed critical t value is 2.093 and 99% two-tailed critical t value is 2.861.
(did you remember to subtract one for degrees of freedom; d.f. = n - 1? Did you note this is a two-tailed test?)
At 95%, 2.24 > 2.093 so we reject the null
At 99%, 2.24 is not > 2.861 so we cannot reject the null (imprecisely, we may say we "accept the null" but truly we fail to reject the null).
In short, we are 95% confident the true pop mean is not 690 but we are not 99% confident: 2.24 standard deviations could occur due to sampling variation with probability of 3.75% (the p value). As the p value is the exact significance level, we could reject the null with exactly 96.25% (1-p) confidence, less but not more.

Note: I like the demonstration that, although, we use the t statistic at 95% confidence at 19 degrees of freedom (two-tailed), when we compute the p value, it is (1-tail) - Good learning for me

Thanks!
Jayanthi

 

[ShaktiRathore]

Yes Jayanthi
We can test the significance of the hypothesis test using p value criterea also besides the normal procedure of comparing tstat with the tcric at a certain CL. Here p value=.035<.05 so we reject null at 5% significance level or at 95% CL whereas at 1% significance .035>.01 therefore we fail to reject the null and accept the null at 99% CL.
Thanks

 

[Dr. Jayanthi Sankaran]

Hi Shakti,

Thanks for the explanation using the p values instead of the t statistics - appreciate it. My question was to do with the fact that we are using p values - one tail, and using t-statistic - two tails. So, my question is (was) do you always use one-tail p-values? and double them for two-tail?

Thanks!
Jayanthi

 

[ShaktiRathore]

Hi
The p value above we calculated is for two tail i think. You calculate two tail test static above look for the corresponding los under 2 tail heading in the table which is nothing but pvalue .
You calculate p value for both sides of the tail for a two tailed test whereas for one side of the tail for one tail test,yes two tail is equivalent to douvling one tail p value. For tstat=2.24 above under two tail the corresponding los would be .035 whereas for one tail it would be .035/2.
Thanks

 

[Dr. Jayanthi Sankaran]

Hi Shakti,

The p value of 0.035 is one-tail. But, I get the gist of what you are saying - p values can be computed two-tailed, also! And that, the two-sided p value = 2*one-sided p value...From the above, we infer that when we use two-tail for t-statistic, we can use one-tail for the p value.

Thanks!
Jayanthi

 

[David Harper CFA FRM]

Hi @Jayanthi Sankaran

No, actually, the p-value of 3.75% in 72.2 is two-tailed which matches the two-tailed test; i.e., you are correct to imply that the test should consistently be one- or two-tailed. Further, there is ultimately no difference on the outcome. Although the p-value is ripe for debate, here is how I simply look at it: the p-value is the area under the rejection region(s). A one-tailed p-value can only be doubled if the distribution is symmetrical, but the student's t is symmetrical so we can. I am going to use Excel because it's easier:

• T.DIST.RT(X = 10/sqrt(20) = 10.236068, df = 19) = 1.8770%; i.e., one-tailed p value is 0.0187
• T.DIST.2T(X = 10.236068, df = 19) = 3.754%; i.e., is twice the one-tailed p value due to symmetry.

I hope that helps!

 

[Dr. Jayanthi Sankaran]

Hi David,

Yes, I agree that the p-value of 3.75% in 72.2 is two-tailed which matches the two tailed test. Somehow, I get a p-value of 4% - even after setting the calculator upto four decimals. I do agree with you, though

Thanks a tonne!
Jayanthi