Final answer:
The test statistic is derived from the sample data and measures how much the observed data differs from what is expected under the null hypothesis, while the critical value is a threshold associated with the significance level that helps determine whether to reject the null hypothesis.
Step-by-step explanation:
In statistical analysis, the test statistic and the critical value are both essential to hypothesis testing, but they serve different purposes. A test statistic is a value calculated from the sample data, using a specific test (like z-test or t-test), which we then compare against the critical value to determine whether to reject the null hypothesis.
For example, in a z-test, the test statistic formula is z = (X - μX) / σX, where X is the sample mean, μX is the population mean under the null hypothesis, and σX is the standard deviation. The test statistic reflects the degree to which the observed sample statistic diverges from what is expected under the null hypothesis. It is not necessarily always positive; it could be negative or positive depending on the data.
The critical value, on the other hand, is a predetermined point on the distribution of the test statistic which defines the threshold for rejecting the null hypothesis. It is based on the significance level (α) chosen by the researcher. If the absolute value of the test statistic is greater than the critical value, the null hypothesis is rejected. In the context of a 5 percent significance level (α = 0.05), critical values correspond to the points beyond which 5 percent of the distribution lies to the extremes.
To summarize, the test statistic is derived from your data and reflects how much your sample departs from the null hypothesis, while the critical value is a cutoff point that is used to decide whether the test statistic is extreme enough to reject the null hypothesis.
Additionally, when comparing the p-value to α, if the p-value is smaller than α, it indicates that the data are unlikely under the null hypothesis, and the null hypothesis is rejected. This is an alternative method to using critical values, where you directly use the probability of observing a test statistic as extreme as, or more extreme than, the observed value.