8.2 Confidence interval, single population mean, standard deviation  (Page 2/9)

Calculators and computers can easily calculate any Student's-t probabilities. The TI-83,83+,84+ have a tcdf function to find the probability for given values of t. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.

For the TI-84+ you can use the invT command on the DISTRibution menu. The invT command works similarly to the invnorm.The invT command requires two inputs: invT(area to the left, degrees of freedom) The output is the t-score that corresponds to the area we specified.

The TI-83 and 83+ do not have the invT command. (The TI-89 has an inverse T command.)

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). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student's-t distribution.) When using 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 (See the Table of Contents 15. Tables ) gives t-scores given the degrees of freedom and the right-tailed probability. The table is very limited. Calculators and computers can easily calculate any Student's-t probabilities.

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

• $T$ ~ $t_{\text{df}}$ where $\text{df}=n-1$ .
• 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 $T$ ~ $t_{\text{19}}$

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

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

The format for the confidence interval is:

$(\overline{x}-\mathrm{EBM},\overline{x}+\mathrm{EBM})$ .

The TI-83, 83+ and 84 calculators have a function that calculates the confidence interval directly. To get to it,
Press STAT
Arrow over to TESTS .
Arrow down to 8:TInterval and press ENTER (or just press 8 ).