Appendix 04: Critical Values for t-Test
- Page ID
- 6640
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Assuming you have calculated texp, there are two approaches to interpreting a t-test. In the first approach you choose a value of α for rejecting the null hypothesis and read the value of t(α,ν) from the table shown below. If texp>t(α,ν), you reject the null hypothesis and accept the alternative hypothesis. In the second approach, you find the row in the table below corresponding to your degrees of freedom and move across the row to find (or estimate) the α corresponding to texp=t(α,ν); this establishes largest value of α for which you can retain the null hypothesis. Finding, for example, that α is 0.10 means that you would retain the null hypothesis at the 90% confidence level, but reject it at the 89% confidence level. The examples in this textbook use the first approach.
| Degrees of Freedom | …a confidence interval of: | 90% | 95% | 98% | 99% |
|---|---|---|---|---|---|
| …an α value of: | 0.10 | 0.05 | 0.02 | 0.01 | |
| 1 | 6.314 | 12.706 | 31.821 | 63.657 | |
| 2 | 2.920 | 4.303 | 6.965 | 9.925 | |
| 3 | 2.353 | 3.182 | 4.541 | 5.841 | |
| 4 | 2.132 | 2.776 | 3.747 | 4.604 | |
| 5 | 2.015 | 2.571 | 3.365 | 4.032 | |
| 6 | 1.943 | 2.447 | 3.143 | 3.707 | |
| 7 | 1.895 | 2.365 | 2.998 | 3.499 | |
| 8 | 1.860 | 2.306 | 2.896 | 3.255 | |
| 9 | 1.833 | 2.262 | 2.821 | 3.250 | |
| 10 | 1.812 | 2.228 | 2.764 | 3.169 | |
| 12 | 1.782 | 2.179 | 2.681 | 3.055 | |
| 14 | 1.761 | 2.145 | 2.624 | 2.977 | |
| 16 | 1.746 | 2.120 | 2.583 | 2.921 | |
| 18 | 1.734 | 2.101 | 2.552 | 2.878 | |
| 20 | 1.725 | 2.086 | 2.528 | 2.845 | |
| 30 | 1.697 | 2.042 | 2.457 | 2.750 | |
| 50 | 1.676 | 2.009 | 2.311 | 2.678 | |
| ∞ | 1.645 | 1.960 | 2.326 | 2.576 | |
The values in this table are for a two-tailed t-test. For a one-tail t-test, divide the α values by 2. For example, the last column has an α value of 0.005 and a confidence interval of 99.5% when conducting a one-tailed t-test.