How to perform a t-test if I can't find the t-score on my t-table?

Question

I'm supposed to perform a hypothesis test with:

sample size = $27$

sample mean = $10.7$

sample standard deviation = $3.6$

Hypothesized mean, $\mu_0=11$

Because the sample size is small and population standard deviation is unknown, I perform a t-test to see if I can reject my null hypothesis.

I find that the t-score is $-0.43$ and:

$\text{p-value} =2p(t_{26} > 0.43)$

This is the t-table I will be provided in tests/exams at my school: https://drive.google.com/file/d/0BxUaD_pbAL2hQzE0dzZtVHhsSlU/view

There is no value close to $0.43$ in the row of $\text{df} = 26$ on the t-table, so what do I do?

Am I supposed to use the z-table instead? Or is using R or some other computational program the only possible way to find the p-value?

Also, regarding the t-table I linked, is this table only for one-tailed t-tests? If so, what do I do if I need to do a two-tailed test (like in this example)?

Answer 1

Also, regarding the t-table I linked, is this table only for one-tailed t-tests?

Directly it's for one-tailed but you can use it with two-tailed tests. I'll explain the one-tailed use and then discuss how to do it for two-tailed tests.

There is no value close to 0.43 in the row of df = 26 on the t-table, so what do I do?

That sort of table is able to give bounds on the p-value. That's sufficient to know whether to reject or not.

$$ \begin{array}{r r r r r|r} \hline t_{0.1}&t_{0.05}&t_{0.025}&t_{0.010}& t_{0.005}&\text{df}\\ \hline \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1.315& 1.706& 2.056& 2.479& 2.779& 26\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \end{array} $$

For example if your t-value was 1.5, which is between 1.315 and 1.706, you know that the one tailed p-value is between 0.1 and 0.05.

With $t=0.433$ you'd only be able to say that the one-tailed p-value was greater than $0.1$ (and for any typical significance level that means you don't reject $H_0$).

what do I do if I need to do a two-tailed test?

Double the one-tailed p-value (or with this table, double the bounds you identify).

In fact the information to double it is already in the formula in your question:

So if you had $|t|=1.5$ you'd say the two-tailed p-value was between $0.2$ and $0.1$.

If you had $|t|=0.433$ you'd say that the two-tailed p-value was greater than $0.2$.

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When your t statistic is between two tabulated values, it is possible to get an approximation to the p-value via interpolation but it's not necessary in this case (and I wouldn't spend the extra time in an exam).