7.2 A single population mean using the student t distribution --  (Page 2/21)

Calculators and computers can easily calculate any Student's t-probabilities.

A probability table for the Student's t-distribution can also be used. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row). When using a t -table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.

A Student's t table gives t -scores given the degrees of freedom and the right-tailed probability. However such a table is very limited. Calculators and computers can easily calculate any Student's t-probabilities.

The notation for the Student's t-distribution (using T as the random variable) is:

• T ~ t _df where df = n – 1.
• For example, if we have a sample of size n = 20 items, then we calculate the degrees of freedom as df = n - 1 = 20 - 1 = 19 and we write the distribution as T ~ t ₁₉ .

If the population standard deviation is not known , the error bound for a population mean is:

• $EBM=\left(t_{\frac{\alpha }{2}}\right)\left(\frac{s}{\sqrt{n}}\right)$ ,
• $t_{\frac{\sigma }{2}}$ is the t -score with area to the right equal to $\frac{\alpha }{2}$ ,
• use df = n – 1 degrees of freedom, and
• s = sample standard deviation.

The format for the confidence interval is:
$(\overline{x}-EBM,\overline{x}+EBM)$ .