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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 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 = t α 2 ( s n )
  • t α 2 is the t-score with area to the right equal to α 2
  • use df = n - 1 degrees of freedom
  • s = sample standard deviation

The format for the confidence interval is:

( x - EBM , x + 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 ).

Suppose you do a study of acupuncture to determine how effective it is in relieving pain. You measure sensory rates for 15 subjects withthe results given below. Use the sample data to construct a 95% confidence interval for the mean sensory rate for the population (assumed normal) fromwhich you took the data.

The solution is shown step-by-step and by using the TI-83, 83+ and 84+ calculators.

  • 8.6
  • 9.4
  • 7.9
  • 6.8
  • 8.3
  • 7.3
  • 9.2
  • 9.6
  • 8.7
  • 11.4
  • 10.3
  • 5.4
  • 8.1
  • 5.5
  • 6.9
  • You can use technology to directly calculate the confidence interval.
  • The first solution is step-by-step (Solution A).
  • The second solution uses the Ti-83+ and Ti-84 calculators (Solution B).

Solution a

To find the confidence interval, you need the sample mean, x , and the EBM.

x = 8.2267 s = 1.6722 n = 15

df = 15 - 1 = 14

CL = 0.95 so α = 1 - CL = 1 - 0.95 = 0.05

α 2 = 0.025 t α 2 = t .025

The area to the right of t .025 is 0.025 and the area to the left of t .025 is 1−0.025=0.975

t α 2 = t .025 = 2.14 using invT(.975,14) on the TI-84+ calculator.

EBM = t α 2 ( s n )

EBM = 2.14 ( 1.6722 15 ) = 0.924

x - EBM = 8.2267 - 0.9240 = 7.3

x + EBM = 8.2267 + 0.9240 = 9.15

The 95% confidence interval is (7.30, 9.15) .

We estimate with 95% confidence that the true population mean sensory rate is between 7.30 and 9.15.

Solution b

Using a function of the TI-83, TI-83+ or TI-84 calculators:

Press STAT and arrow over to TESTS .
Arrow down to 8:TInterval and press ENTER (or you can just press 8 ). Arrow to Data and press ENTER .
Arrow down to List and enter the list name where you put the data.
Arrow down to Freq and enter 1.
Arrow down to C-level and enter .95
Arrow down to Calculate and press ENTER .
The 95% confidence interval is (7.3006, 9.1527)

When calculating the error bound, a probability table for the Student's-t distribution can also be used to find the value of t. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row); the t-score is found where the row and column intersect in the table.

**With contributions from Roberta Bloom

Practice Key Terms 7

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Source:  OpenStax, Elementary statistics. OpenStax CNX. Dec 30, 2013 Download for free at http://cnx.org/content/col10966/1.4
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