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In practice, we rarely know the population standard deviation . In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Goset (1876–1937) of the Guinness brewery in Dublin, Ireland ran into this problem. His experiments with hops and barley produced very few samples. Just replacing σ with s did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size. This problem led him to "discover" what is called the Student's t-distribution . The name comes from the fact that Gosset wrote under the pen name "Student."
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.
If you draw a simple random sample of size n from a population with mean μ and unknown population standard deviation σ and calculate the t -score t = $\frac{\stackrel{-}{x}\u2013\mu}{\left(\frac{s}{\sqrt{n}}\right)}$ , then the t -scores follow a Student's t-distribution with n – 1 degrees of freedom . The t -score has the same interpretation as the z -score . It measures how far in standard deviation units $\stackrel{-}{x}$ is from its mean μ . For each sample size n , there is a different Student's t-distribution.
The degrees of freedom , n – 1 , come from the calculation of the sample standard deviation s . Remember when we first calculated a sample standard deviation we divided the sum of the squared deviations by n − 1, but we used n deviations $(x\u2013\stackrel{-}{x}\text{values})$ to calculate s . Because the sum of the deviations is zero, we can find the last deviation once we know the other n – 1 deviations. The other n – 1 deviations can change or vary freely. We call the number n – 1 the degrees of freedom (df) in recognition that one is lost in the calculations. The effect of losing a degree of freedom is that the t-value increases and the confidence interval increases in width.
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