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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.