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16.4: Critical Values for t-Test

  • Page ID
    220801
  • Assuming we have calculated texp, there are two approaches to interpreting a t-test. In the first approach we choose a value of \(\alpha\) for rejecting the null hypothesis and read the value of \(t(\alpha,\nu)\) from the table below. If \(t_\text{exp} > t(\alpha,\nu)\), we reject the null hypothesis and accept the alternative hypothesis. In the second approach, we find the row in the table below that corresponds to the available degrees of freedom and move across the row to find (or estimate) the a that corresponds to \(t_\text{exp} = t(\alpha,\nu)\); this establishes largest value of \(\alpha\) for which we can retain the null hypothesis. Finding, for example, that \(\alpha\) is 0.10 means that we 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.

    Table \(\PageIndex{1}\): Critical Values of t for the t-Test
    Values of t for…        
    …a confidence interval of: 90% 95% 98% 99%
    …an \(\alpha\) value of: 0.10 0.05 0.02 0.01
    Degrees of Freedom        
    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
    \(\infty\) 1.645 1.960 2.326 2.576

    The values in this table are for a two-tailed t-test. For a one-tailed test, divide the \(\alpha\) values by 2. For example, the last column has an \(\alpha\) value of 0.005 and a confidence interval of 99.5% when conducting a one-tailed t-test.

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