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Transcript
Student’s T Distribution – TI 83/84
Student’s T distribution gives substantially greater accuracy when looking at the distribution of the
means of small samples from a population. It’s actual use is to evaluate the precision of tests based on a
single small sample. There are, as usual, conditions that have to be checked to make sure that the use
of the distribution is valid. These conditions are:
1. The sample elements are randomly chosen from the population.
2. The sample constitutes less than 10% of the population.
3. The variable being tested is distributed approximately normally in the population
a)
For n < 15, the approximation to Normal should be fairly close.
b)
For 15< n < 40, the distribution should be unimodal and symmetric.
c)
For n > 50, the distribution is fairly safe, even if the data are skewed.
There are two kinds of test that can be done: a confidence interval and a hypothesis test. The
confidence interval is slightly easier to evaluate on the TI 83.
CONFIDENCE INTERVAL
Again, I have used the convention that if a single calculator button is required, the label on the button is
highlighted in gray, e.g. ENTER.
It is assumed here that the data have already been entered into one of the calculator’s lists and it is
known which list contains the data. If instruction for that procedure is needed, see the article on Mean
and Standard Deviation on the TI-83.
To start the calculation of the confidence interval, press the key labeled STATS. The screen will display:
Press the left arrow key once, or the right arrow twice to move
the highlighted area in the top command ribbon to the word
TESTS. Do NOT press the ENTER key at this point.
A sub-menu will be displayed :
The needed command is actually number 8 on this menu and
does not show at this point. To reach the desired prompt,
press the down arrow until the prompt 8: TInterval… shows,
with the numeral 8: highlighted and then press the ENTER key.
An alternative procedure is to simply press the 8 key when this
menu is displayed. That is simpler, but requires remembering
the command.
Either procedure calls up the screen in which the confidence interval is calculated:
The first line is simply an identifier to tell the user that they are
about to calculate a confidence interval using the T distribution.
The second line tells tell calculator how to access information.
The highlighted area on the right side of the line actually
contains the blinking word Stats. That access mode requires
that the user has already calculated the mean and standard
deviation of the data (or they are given in a problem) and plans to enter them manually. Since the
procedure here assumes that the data have been typed into a list, the Data mode will be used. If Stats is
highlighted, press the left arrow key to highlight the word Data. Pressing the ENTER key will change the
display to:
The word Data is blinking inside the highlighted area on the
second Line. The mode can be changed back to Stats, if desired.
Instead, press the down arrow. The cursor will move to the
third line where the name of the list that contains the data is to
be entered. Pressing the down arrow again moves the cursor to
the fourth line. Freq: is to be left at 1. The purpose of the Freq
command is to allow the user to enter a single data item and
then tell the calculator that there is more than one instance of that number in the data set. Entering the
entire set into the list makes the use of the Freq: command unnecessary.
The fifth line has the prompt C-Level: . This is the desired confidence level of the interval. in decimal
form. The default is whatever confidence level was entered the last time the procedure was used. (That
fact can be used as a reminder in case the number is forgotten.) It should always be checked to be sure
that the confidence level is what is currently desired.
After checking the confidence level, pressing the down arrow causes the word Calculate to be
highlighted. Pressing the ENTER key tells the calculator to perform the calculation. This takes about 5
seconds, and then a screen similar to the one below is displayed.
The second line (first line of numbers) is the desired confidence
interval. 𝑥̅ is the mean of the numbers in the list, Sx is the
standard deviation of the numbers in the list, and n is the
sample size (number of items in the calculator’s list).
As an example problem, calculate a 96% confidence interval for the data set:
415
398
402
407
386
400
391
410
The confidence interval is from 392.5 to 409.7
HYPOTHESIS TEST
Again, it is assumed that the data have been entered into one of the calculator’s lists.
However, before beginning this calculation it is necessary to choose both null and alternative
hypotheses, since information from both must be entered into the calculator.
To start the calculation of the hypothesis test, press the key labeled STATS. As it did for the confidence
interval, the screen will display:
Press the left arrow key once, or the right arrow twice to move
the highlighted area in the top command ribbon to the word
TESTS. Do NOT press the ENTER key at this point or the
calculator will start on the first menu item in the sub-menu.
The sub-menu will be displayed :
The needed command is actually number 2 on this menu and
does not show at this point. To reach the desired prompt,
press the down arrow once so that the prompt 2: T-Test…
is highlighted and then press the ENTER key.
An alternative, simpler, procedure is to press the 2
key when this menu is displayed.
In either case, the next screen is:
As with the Confidence Interval, the first line displays the words
Data Stats. One of these words will be highlighted, depending
on which choice was used the last time this test was performed
on the calculator. Since the data are in a list, the word Data
should be highlighted and the ENTER key pressed. The cursor
moves to the next line where the value for μo is entered. This is
the number which was chosen for the null hypothesis. The next
line tells the calculator which list contains the sample data. Frequency remains at 1. The final entry line
tells the calculator whether this is a two-tail test (≠ μ0), a lower tail test (< μ0), or an upper tail test (> μ0).
The last line gives the user a choice of what the calculator should do. Using the data in the example
above, and hypotheses H0 = 400, Ha ≠ 400, the CALCULATE command displays the screen:
The first line below the T-Test label repeats what alternative
hypothesis was chosen, here a two tail test. The second line
gives t, the distance in standard errors of the sample mean
from the assumed population mean, which was entered in the
previous screen. The next line is the p-value, the probability
that this distance could occur by random chance. Note that
there was no place to enter an α value. Instead, the calculator
gives a p-value which the user can compare with their internalized value for α. They can then choose to
reject the null hypothesis, or not. The last three lines are the mean, standard deviation and sample size.
The numbers shown on the screens are the numbers generated for the example given at the end of the
section on Confidence Intervals.
For the example, the null hypotheis was that μo =400 and the alternative hypothesis was that
μ0 ≠ 400, a two tail test. The distance between the sample mean (𝑥̅ = 401.1) and the assumed
population mean of 400 was .33 standard errors and the probability of that distance occuring by random
chance was .75, or 75%. The other numbers are calculated values for the sample data.
With a p-value of .75, obviously the null hypothesis cannot be rejected.
If the DRAW option had been chosen instead of CALCULATE, the resulting screen would be:
As with the calculate option, the value of t and the p-value
are displayed, but the picture gives a picture of the areas inside
and outside the region determined by the calculated value of t.