Hypothesis Testing: Comparing Sample Values To Critical Values In T-Tests And Chi-Squared Tests

The value calculated from the sample which is used to perform the test

The test statistic

The value calculated from the sample which is used to perform the test depends on the type of test being performed. In general, the value calculated from the sample is compared to a critical value, which is determined by the distribution that the data follows or by a pre-determined alpha level. If the calculated value exceeds the critical value, the null hypothesis is rejected and the alternative hypothesis is accepted. If the calculated value is less than or equal to the critical value, then the null hypothesis is not rejected.

For example, in the t-test for two independent samples, the value calculated from the sample is the t-statistic, which is calculated by taking the difference between the means of the two samples and dividing it by the standard error of the difference. This t-statistic is then compared to a critical value from the t-distribution with degrees of freedom equal to the sample sizes minus two.

In the chi-squared test for goodness of fit, the value calculated from the sample is the chi-squared statistic, which is calculated by summing the squared differences between the observed frequencies and the expected frequencies, divided by the expected frequencies. This chi-squared statistic is then compared to the critical value from the chi-squared distribution with degrees of freedom equal to the number of categories minus one.

In both cases, the value calculated from the sample is a measure of how far the sample data deviates from what would be expected under the null hypothesis. By comparing this value to a critical value, we can determine whether the deviation is significant enough to reject the null hypothesis.

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