With graphing calculators and computers, the practice now is to use the Student’s t-distribution whenever sis used as an estimate for σ. Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and only used the Student’s t-distribution only for sample sizes of at most 30. The name comes from the fact that Gosset wrote under the pen name “Student.” This problem led him to “discover” what is called the Student’s t-distribution. He realized that he could not use a normal distribution for the calculation he found that the actual distribution depends on the sample size. Just replacing σ with s did not produce accurate results when he tried to calculate a confidence interval. His experiments with hops and barley produced very few samples. Goset (1876–1937) of the Guinness brewery in Dublin, Ireland ran into this problem. A small sample size caused inaccuracies in the confidence interval. However, statisticians ran into problems when the sample size was small. They used the sample standard deviation s as an estimate for σand proceeded as before to calculate a confidence interval with close enough results. In the past, when the sample size was large, this did not present a problem to statisticians. So, when getting df2, the formula will be as follows: df2 n k. And this is because the degrees of freedom will be two in this case. When it comes to 3-groups Anova, the calculation will be different from 2-group Anova. In practice, we rarely know the population standard deviation. In this case, the degrees of freedom for 2-group Anova 1.
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