What is the difference between a Critical value and a Test statistic and between a Z-score and a T-statistic?

Critical values for a test of hypothesis depend upon a test statistic, which is specific to the type of test, and the significance level, α, which defines the sensitivity of the test. A value of α = 0.05 implies that the null hypothesis is rejected 5 % of the time when it is in fact true. The choice of α is somewhat arbitrary, although in practice values of 0.1, 0.05, and 0.01 are common. Critical values are essentially cut-off values that define regions where the test statistic is unlikely to lie; for example, a region where the critical value is exceeded with probability α if the null hypothesis is true. The null hypothesis is rejected if the test statistic lies within this region which is often referred to as the rejection region(s).

The Z-score allows you to decide if your sample is different from the population mean. In order to use z, you must know four things: The population mean, the population standard deviation, the sample mean, and the sample size. Usually in statistics, you don’t know anything about a population, so instead of a Z- score you use a T-Test with a T -Statistic. The major difference between using a Z-score and a T-statistic is used when you have to estimate the population standard deviation. The T-test is also used if you have a small sample size e.g., less than 30.