critical or calculated t

[hellohi]

dear @David Harper CFA FRM

in your video related to quantitative book, Miller chapter 7 and slide number 8, you mentioned t critical and you said it is the same as calculated t,,,,, I think the critical t is the tabulated not the calculated....and what t we should to use to calculate confidence interval? I think the critical from the table.

thanks
Nabil

 

[David Harper CFA FRM]

Hi @hellohi

I may have mispoke (I don't have current time to check, sorry) or maybe the context makes it difficult. But, I agree with you. Critical-t is what I also call a "lookup t." Yes, we use the critical-t for the confidence interval because the critical-t is, by definition, standardized. Here's what I mean. Say we observe sample mean of 10.0 and sample variance of 4.0 for n = 20 observations. 95% CI is given by 10.0 +/- sqrt(4/20)*2.093, where 2.093 = T.INV.2T(5%, 9 df) such that 2.093 is the critical-t or lookup-t being standardized. This CI = sample mean +/- SE*(critical-t) does not contain the "computed t" (aka, test statistic). That's confidence intervals. On the other hand, we can retrieve a "computed t" or "test statistic" but, whereas the CI requires a confidence level the computed t requires a null hypothesis. In this case, for example, maybe null is H(0): true mean = 12.0. In this case, computed t (test statsitic) = abs(12-10)/SE = 4.47. Now we compare this the critical t (lookup) t of 2.093 and see that it's greater (i.e., falls in the rejection region). All of the same elements are used in both approaches! I hope that helps, thanks!

 

[hellohi]

Hi @hellohi

I may have mispoke (I don't have current time to check, sorry) or maybe the context makes it difficult. But, I agree with you. Critical-t is what I also call a "lookup t." Yes, we use the critical-t for the confidence interval because the critical-t is, by definition, standardized. Here's what I mean. Say we observe sample mean of 10.0 and sample variance of 4.0 for n = 20 observations. 95% CI is given by 10.0 +/- sqrt(4/20)*2.093, where 2.093 = T.INV.2T(5%, 9 df) such that 2.093 is the critical-t or lookup-t being standardized. This CI = sample mean +/- SE*(critical-t) does not contain the "computed t" (aka, test statistic). That's confidence intervals. On the other hand, we can retrieve a "computed t" or "test statistic" but, whereas the CI requires a confidence level the computed t requires a null hypothesis. In this case, for example, maybe null is H(0): true mean = 12.0. In this case, computed t (test statsitic) = abs(12-10)/SE = 4.47. Now we compare this the critical t (lookup) t of 2.093 and see that it's greater (i.e., falls in the rejection region). All of the same elements are used in both approaches! I hope that helps, thanks!

thanks dear david