To find the area between two numbers a and b under the chi-square curve, use the χ 2 cdf( function.

Transcription

1 Chapter 11 Chi-Square Tests Computing Chi-Square Distribution Probabilities Chi-square probabilities can be computed by using the χ 2 cdf( function. This function can be accessed by pressing ( ) and looking in the DISTR menu. Compute Cumulative Chi-square Probabilities The χ 2 cdf( function stands for chi-square cumulative density function and gives the probability of getting a x 2 value that falls within an interval of values from the chi-square distribution for the specified degrees of freedom. There are three possibilities: Finding the probability that a number will fall between two values under the chi-square distribution. Finding the probability that a number will fall to the left of a value under the chi-square distribution (in the left tail). Finding the probability that a number will fall to the right of a value under the chi-square distribution (in the right tail). The syntax for the χ 2 cdf( function is χ 2 cdf(a, b, df), where a is the lower bound of the interval, b is the upper bound of the interval, and df is the degrees of freedom. Finding the Area Between Two Values To find the area between two numbers a and b under the chi-square curve, use the χ 2 cdf( function. Find the probability of getting a value between 5.14 and 7.28 under the chi-square curve with 8 degrees of freedom.

2 Press ( ). Select 8: χ 2 cdf( from the DISTR menu. At the lower prompt, type At the upper prompt, type At the lower prompt, type 8. Highlight Paste and press twice. P(5.14 < χ 2 < 7.28) = Finding the Area to the Left of a Value (in the Left Tail) To find the area to the left of b (in the left tail) under the chi-square curve, use the χ 2 cdf( function. The lower bound should be negative infinity (- ). The problem is that the TI-84 calculator does not have a built-in key for negative infinity (- ). Thus, the value -1E99 is used, which represents a negative number that lies far to the left on the number line. The letter E stands for scientific notation and it is located above the comma key (press ). Find the probability of getting a value less than 20 under the Chi-square curve with 17 degrees of freedom. Press ( ). Select 8: χ 2 cdf( from the DISTR menu. At the lower prompt, type -1E99. At the upper prompt, type 20. At the lower prompt, type 17. Highlight Paste and press twice. P(χ 2 < 20) =

3 Finding the Area to the Right of a Value (in the Right Tail) To find the area to right of a (in the right tail) under the chi-square curve, use the χ 2 cdf( function. The upper bound should be positive infinity (+ ). Since the TI-84 calculator does not have a built-in key for negative infinity (+ ), use the value 1E99, which represents a positive number that lies far to the right on the number line. The letter E stands for scientific notation and it is located above the comma key (press ). Find the probability of getting a value greater than under the standard chi-square curve with 25 degrees of freedom. Select 8: χ 2 cdf( from the DISTR menu. At the lower prompt, type At the upper prompt, type 1E99. At the lower prompt, type 25. Highlight Paste and press twice. P(χ 2 > 31.08) = Graph the Chi-square Probability Density Function The function χ 2 pdf( stands for chi-square probability density function and does not actually generate a probability, since it applies to a single x value in a continuous distribution and that probability is always zero. The main use of this command is to draw the chi-square curve. The syntax for the function is χ 2 pdf(x, df), where df is the degrees of freedom. The following sequence of commands will draw the chi-square curve with 7 degrees of freedom. Press. If a function is already entered in Y 1, press. Select 7: χ 2 pdf( from the DISTR menu. At the x value prompt, press. At the df prompt, type 7. Highlight Paste and press. 3

4 To see the graph, use these window settings Xmin = 0 Xmax = 14 Xscl = 1 Ymin = Ymax = 0.20 Yscl = 0.05 This command may be used to draw any chi-square distribution curve with any degrees of freedom, although it may be necessary to adjust the window settings from those listed above. Shade the Chi-square Probability Density Function When calculating the probability of an area under the chisquare curve, it is often helpful to shade the area. The syntax for the TI-84 Plus command to do this is Shade χ 2 (a, b, df). To shade the area under the standard chi-square curve with 8 degrees of freedom for P(5.14 < χ 2 < 7.28), begin by turning off all other plots and graphs ( and ). Press and enter the values shown in the screenshot shown to the left. Select 3: Shadeχ 2 ( from the DRAW menu. At the lower prompt, type At the upper prompt, type At the lower prompt, type 8. Highlight Draw and press. Notice that the area of the shaded region is also shown on the graph and it is the same value calculated from the χ 2 cdf( command. Thus, the ShadeNorm( is an alternative command for χ 2 cdf(, with the added benefit of the shading of the area. 4

5 A Goodness-of-Fit Test A goodness-of-fit test is used to make a test of hypothesis about experiments with more than two possible outcomes (categories). These are called multinomial experiments. The frequencies (counts) of each possible outcome obtained from the experiment are called the observed frequencies (counts). A goodness-of-fit test tests the sum of the squared differences between the observed frequencies and the expected frequencies (np i ). This statistic follows a chi-square distribution. The sample size should be large enough so that the expected frequency (np i ) for each category is at least 5. The TI-84 Plus command for the goodness-of-fit test is χ 2 GOF-Test. This test checks to see how much the sample data differs from a specified population distribution. This function requires that both the observed and expected frequencies are put in lists. The degrees of freedom, df, are the number of outcomes (categories) minus 1. This function can be accessed by pressing looking under TESTS. Example 1: and The following table lists the frequency distribution of 90 rolls of a die. Test at the 5% significance level whether the null hypothesis that the given die is fair is true. (Note: if the die is fair, we would expect to roll each number 15 times.) Outcome of roll Observed Frequency Expected Frequency Enter the observed frequencies into list L1 and the expected frequencies into list L2, as shown to the right. There are 6 possible outcomes, so the degrees of freedom, df, is 6 1 = 5. Press and select D: χ 2 GOF-Test from the TESTS menu. At the Observed prompt, press. At the Expected prompt, press. At the df prompt, type 5. Highlight Calculate and press ENTER. 5

6 The output for the χ 2 GOF-Test shows the: chi-square value: χ 2 = p-value = degrees of freedom: df = 5 CNTRB = { , 2.4, 0,.6, ,.6} Note: CNTRB= provides a list of the contributions of each category to the overall value of χ 2. This would be the values in the ( roll of 2 had the largest contribution. ) column. Notice that a Since the p-value is greater than 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that the dice is unfair. Contingency Tables When measuring the relationship between two categorical variables, one of the most important tools for analyzing the results is a two-way classification table, also known as a contingency table. A Test of Independence The contingency table can be used to see if the variables are independent, by comparing observed frequencies with the frequencies that would be expected from such a sample if they were independent. A chi-square (χ 2 ) test-statistic can be computed from the observed and expected frequencies. The TI- 84 function χ 2 -Test is used for the Test of Independence and can be accessed by pressing and looking in the TESTS menu. The χ 2 -Test function works differently than the other tests on this menu. It requires you to enter the observed frequencies into a matrix. The expected frequencies are then computed and stored automatically in another matrix. Example: School Referendum A random sample of 300 adults was selected and asked if they were in favor of the new school referendum. The two-way classification table of the responses of the adults is presented in the table to the right. In Favor Against No Opinion Men Women Test at a 5% significance level whether or not Gender and Opinion are independent. 6

7 First the data has to be stored in a matrix. Select 1: [A] from the EDIT menu. Type 2 and then type 3 for the dimensions. The 2 represents the number of rows in the table. The 3 represents the number of columns in the table. Enter the data values into the matrix as they appear in the table. (Use the arrow keys or to move from cell to cell). Press and select C: χ 2 -Test from the TESTS menu. The χ 2 -Test screen shows: That the Observed values are in matrix A That the Expected values will be put in matrix B Highlight Calculate and press ENTER. The output for the χ 2 -Test shows the: test statistic: χ 2 = p-value: p = degrees of freedom: df = 2 Since the p-value of.016 is less than the significance level of.05, we reject the null hypothesis that row and column variables are independent. We have statistically significant evidence that gender and opinion concerning the school referendum are dependent for all adults. The expected frequencies can be found in Matrix B. Under the NAMES menu, highlight 2: [B], and press. In the above example, the Calculate option was chosen. If the Draw option is chosen, the calculator will draw the curve and state the χ 2 value and the p-value. 7

8 A Test of Homogeneity A test of homogeneity is a test to determine if two (or more) populations are homogeneous (similar) with regard to the distribution of a certain variable. The procedure to perform this test on the TI-84 Plus is identical to performing a test of independence. Refer to the instructions given above. Inferences About the Population Variance In the same way that the population mean and population proportion are tested, so is the population variance. This is often in response to a desire to control the consistency of a value. If the population from which the sample is taken is approximately normally distributed, then the sample variance has a chi-square distribution with n - 1 degrees of freedom. The TI-84 Plus does not have a built-in function to generate confidence intervals or conduct tests of hypothesis about the population variance. 8