9.2 Confidence interval, single population mean, standard deviation

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 thesample standard deviation $s$ as an estimate for $\sigma$ 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 theconfidence interval.

William S. Gossett (1876-1937) of the Guinness brewery in Dublin, Ireland ran into this problem. His experiments with hops and barley produced very fewsamples. Just replacing $\sigma$ with $s$ did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distributionfor 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 for sample sizes of at most 30.With the common use of graphing calculators and computers, the practice is to use the Student's-t distribution whenever $s$ is used as an estimate for $\sigma$ .

If you draw a simple random sample of size $n$ from a population that has approximately a normal distribution with mean $\mu$ and unknown population standard deviation $\sigma$ and calculate the t-score $\mathrm{t}=\frac{\overline{\mathrm{x}}-\mu }{\left(\frac{s}{\sqrt{n}}\right)}$ , 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 $\overline{x}$ is from its mean $\mu$ . 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 $s$ . In Chapter 2, we used $n$ deviations $(x-\overline{x}\text{values})$ to calculate $s$ . Because the sum of the deviations is 0, we can find the last deviation once we know theother $n-1$ deviations. The other $n-1$ deviations can change or vary freely. We call the number $n-1$ the degrees of freedom (df).

Properties of the student's-t distribution

• The graph for the Student's-t distribution is similar to the Standard Normal curve.
• The mean for the Student's-t distribution is 0 and the distribution is symmetric about 0.
• The Student's-t distribution has more probability in its tails than the Standard Normal distribution because the spread of the t distribution is greater than the spread of the Standard Normal. So the graph of the Student's-t distribution will be thicker in the tails and shorter in the center than the graph of the Standard Normal distribution.
• The exact shape of the Student's-t distribution depends on the "degrees of freedom". As the degrees of freedom increases, the graph Student's-t distribution becomes more like the graph of the Standard Normal distribution.
• The underlying population of individual observations is assumed to be normally distributed with unknown population mean $\mu$ and unknown population standard deviation $\sigma$ . The size of the underlying population is generally not relevant unless it is very small. If it is bell shaped (normal) then the assumption is met and doesn't need discussion. Random sampling is assumed but it is a completely separate assumption from normality.