'manual' t-test gives different result for same t-value

Hello, maybe someon can help. Why do I get a different result for the p-value if I enter the tstat-value in to the tpdf-distribution?

mu=500;
sigma=25;

Sample1 = [486,520,497,510,515,521,512,490,520,496];

t_val = sqrt(length(Sample1))*(mean(Sample1)   - mu)/sqrt(var(Sample1));

disp(['t-value = ',num2str(t_val)])

t-value = 1.598

p_val = tpdf(t_val,9);

disp(['p-value = ',num2str(p_val)])

p-value = 0.1113

[~,p,~,stats]=ttest(Sample1,mu)

p = 0.1445

stats = struct with fields:
    tstat: 1.5980
       df: 9
       sd: 13.2585

disp(['p-value = ',num2str(p)])

p-value = 0.14451

Shouldn't the results be the same, since the t-value is the same?

Answer 1

[John D'Errico]

I think you misunderstand what a pdf is, when to use a cdf versus a pdf, or perhaps how the t-test works. Maybe all of those are issues here.

mu=500;
sigma=25;
Sample1 = [486,520,497,510,515,521,512,490,520,496];
t_val = sqrt(length(Sample1))*(mean(Sample1)   - mu)/sqrt(var(Sample1))

t_val = 1.5980

So we have tval.

But now you did this:

p_val = tpdf(t_val,9)

p_val = 0.1113

I'm not certain what you think that does. Do you think that sets some p value for the ttest? The ttest function does not know anything about the variable p_val, nor does it care. Hmm. I see. It looks like you expected p_val to be the same thing as the number p as returned from ttest?

[~,p,~,stats]=ttest(Sample1,mu)

p = 0.1445

stats = struct with fields:
    tstat: 1.5980
       df: 9
       sd: 13.2585

The variable p that is returned from ttest comes from an integral of the t pdf. That 0.1445 is derived from the tcdf function. We can get that value as:

tcdf(stats.tstat,9,'upper')*2

ans = 0.1445

So a two sided t-test, since the t-distribution is symmetrical.

Using the pdf there as you did is simply incorrect.