For the t-table you need to have a separate set of data for any DF value, hence the Z-Table is more detailed and more accurate than the t-table. In the past, people used tables to calculate the cumulative probability. The t-distribution limit at infinity degrees of freedom is the normal distribution. When the sample size is greater than 30, the t-distribution is very similar to the normal distribution. The degrees of freedom equal sample size minus one. I guess the reason for the confusion is historical. The t-test is always the correct test when you estimate the sample standard deviation. When the sample size is larger than 30, should I use the z-test?You should use the t-test! Usually, we don't have the population standard deviation, so we use the t-test. (the standard deviation of the standard deviation statistic) The t-distribution has heavier tails ( Leptokurtic Kurtosis) than the normal distribution to compensate for the higher uncertainty because we estimate the standard deviation. When to use the z-test vs t-test?When you know the population standard deviation you should use the z-test, when you estimate the sample standard deviation you should use the t-test.
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