The Chi-Square Goodness-of-Fit Test, Equal Proportions

Transcription

1 Chapter 11 Chi-Square Tests 1 Chi-Square Tests Chapter 11 The Chi-Square Goodness-of-Fit Test, Equal Proportions A hospital wants to know if the proportion of births are the same for each day of the week. In a random sample of 100 births, the number on each day of the week is shown below: Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Number Births To enter the raw data, in column C1 enter 32 Monday rows, 16 Tuesday rows, 7 Wednesday rows, and so on, until you have 100 rows corresponding to each birth.

2 Chapter 11 Chi-Square Tests 2 To perform the Goodness of Fit Test to determine if the proportion of each day of the week is the same, click on Stat Tables Chi-square Goodness of Fit Test (One Variable). Click on the circle next to Categorical data and select the data in column C1. Select the circle to test Equal proportions. Click OK. Two graphs will be displayed. The first is a Chart of Contribution to the Chi-Square Value by Category. These are the values of ( O E) E graph by clicking on the X in the right corner of the window. 2 for each category. Close this

3 Chapter 11 Chi-Square Tests 3 The second graph that is displayed is the Chart of Observed and Expected Values for each category. Close this graph by clicking on the X in the right corner of the window. The results of the Chi-Square Goodness of Fit Test will be displayed in the Session 2 window. is 36.5, and the p-value is Therefore, for any level of the correct decision would be to reject the null hypothesis.,

4 Chapter 11 Chi-Square Tests 4 The summarized data may also be entered into Minitab. In a Minitab worksheet, enter each Day of the Week in column C1 and the Number of Births for each day in column C2. To perform the Goodness of Fit Test that the proportion of each day of the week is the same, click on Stat Tables Chi-square Goodness of Fit Test (One Variable). Click on the circle next to Observed counts and select the data in column C2. In the Category name field, select the data in column C1. Select the circle to test Equal proportions. Click OK. The two graphs will be displayed, and the results of the Chi-Square Goodness of Fit Test will be displayed in the Session window.

5 Chapter 11 Chi-Square Tests 5 The Chi-Square Goodness-of-Fit Test, Unequal Proportions Using the above data for the number of births on each day of the week, assume the hospital instead wants to test that the proportion of births on Mondays is twice that of every other day of the week. This means that the proportion of births on Mondays is 0.25 and each of the other days is The data may be entered in either raw format or summary format, as described above. For this example, the summary format will be shown. To perform the Goodness of Fit Test for these proportions, click on Stat Tables Chi-square Goodness of Fit Test (One Variable). Click on the circle next to Observed counts and select the data in column C2. In the Category name field, select the data in column C1. Select the circle to test Specific proportions. In the pulldown, select the option to Input Constants. A table will appear on the right side of the window, with each of the Category Names displayed in the first column. In the proportions column, enter the test proportions as described above. Click OK.

6 Chapter 11 Chi-Square Tests 6 The two graphs will be displayed, and the results of the Chi-Square Goodness of Fit Test 2 will be displayed in the Session window. is 15.04, and the p-value is Therefore, for any level of 0.020, the correct decision would be to reject the null hypothesis.

7 Chapter 11 Chi-Square Tests 7 The Chi-Square Test of Independence or Homogeneity A random sample of 100 adult Americans was selected and they were asked if they favor higher taxes in return for government-funded healthcare. The data is shown below. In Favor Against No Opinion Males Females To enter the raw data, label column C1 Gender and label column C2 Opinion. Enter the 100 rows of data. There should be 16 rows of data with a gender of Male and an Opinion of In Favor, 26 rows of data with a Gender of Female and an Opinion of In Favor, and so forth.

8 Chapter 11 Chi-Square Tests 8 To perform the Chi-Square Independence test, click on Stat Tables Cross Tabulation and Chi-square. In the Categorical Variables fields, select column C1 for rows and column C2 for columns. Click the Chi-Square button. Select the checkbox next to Chi-square analysis. Click OK. Click OK on the Cross Tabulation and Chi-Square window.

9 Chapter 11 Chi-Square Tests 9 The results of the Chi-Square Test will be displayed in the Session window. 2 is 5.630, and the p-value is Therefore, for any level of 0.060, the correct decision would be to reject the null hypothesis.

10 Chapter 11 Chi-Square Tests 10 The summarized data may also be entered into Minitab. In a Minitab worksheet, enter the values of each cell from the table into columns C1-C3. There is no need to label the columns or the rows. To perform the chi-square independence test, click on Stat Tables Chi-square Test (Two-Way Table in Worksheet). In the Columns containing the table field, select columns C1, C2, and C3. Click OK.

11 Chapter 11 Chi-Square Tests 11 The results of the Chi-Square Test will be displayed in the Session window. 2 is 5.630, and the p-value is Therefore, for any level of 0.060, the correct decision would be to reject the null hypothesis.

12 Chapter 11 Chi-Square Tests 12 Inferences Concerning the Population Variance A machine that fills packages of pretzels is set up so that the average net weight of a package is 32 ounces with a variance of.02 square ounces. An acceptable variance is.02 sq. ounces or less. A sample of 35 packages from the production line gave a sample variance of The weight of each package in the sample is shown below: Confidence Interval for the Population Variance To analyze the data in raw format, enter the data points into column C1 of a Minitab worksheet.

13 Chapter 11 Chi-Square Tests 13 To construct a 99% confidence interval for the variance, click on Stat Basic Statistics 1 Variance. In the Data pulldown, select the option Samples in columns. In the Columns field, select the data in column C1. Select the Options button. Enter 99 in the Confidence Level field. Make sure the Alternative field has the not equal option selected. Click OK. Click OK on the 1 Variance window.

14 Chapter 11 Chi-Square Tests 14 The results will be displayed in the Session window. The 99% confidence interval for the variance (assuming a normal distribution) is (0.0182, ).

15 Chapter 11 Chi-Square Tests 15 In the above example, the data was entered in raw format. If instead you have the summary data, click on Stat Basic Statistics 1 Variance. In the pulldown, select Sample Variance. Enter a Sample size of 35 and a Sample variance of Select the Options button. Enter 99 in the Confidence Level field. Make sure the Alternative field has the not equal option selected. Click OK. Click OK on the 1 Variance window. The results will be displayed in the Session window. The 99% confidence interval for the variance (assuming a normal distribution) is (0.0182, ).

16 Chapter 11 Chi-Square Tests 16 Hypothesis Test for the Population Variance Using the example from above regarding the weight of the pretzel packages, assume that if the variance is greater than.02, the machine filling the bags needs to be adjusted. Perform the hypothesis test to determine if the machine needs to be adjusted. To analyze the data in raw format, enter the data points into column C1 of a Minitab worksheet.

17 Chapter 11 Chi-Square Tests 17 To test the hypothesis 2 H O : 0.02 H 2 1 : 0.02 click on Stat Basic Statistics 1 Variance. In the Data pulldown, select Samples in columns. In the Columns field, select the data in column C1. Select the checkbox to Perform hypothesis test. In the pulldown, select the Hypothesized variance and enter a Value of Select the Options button. The Confidence Level field will be defaulted to 95.0, for a 95% confidence interval. This will not need to be edited this unless you would also like to perform a different confidence interval at this time. This will not influence the results of the hypothesis test. Since the alternative hypothesis is greater than for this example, make sure the Alternative field has the greater than option selected. Click OK. Click OK on the 1 Variance window.

18 Chapter 11 Chi-Square Tests 18 The results will be displayed in the Session window. In the test of Sigma-squared = vs > 0.02, is 53.70, and the p-value is Therefore, for any level of 0.017, the correct decision would be to not reject the null hypothesis.

19 Chapter 11 Chi-Square Tests 19 In the above example, the data was entered in raw format. If instead you have the summary data, click on Stat Basic Statistics 1 Variance. In the pulldown, select Sample Variance. Enter a Sample size of 35 and a Sample variance of Select the checkbox to Perform hypothesis test. In the pulldown, select the Hypothesized variance and enter a Value of Select the Options button. The Confidence Level field will be defaulted to 95.0, for a 95% confidence interval. This will not need to be edited this unless you would also like to perform a different confidence interval at this time. This will not influence the results of the hypothesis test. Since the alternative hypothesis is greater than for this example, make sure the Alternative field has the greater than option selected. Click OK. Click OK on the 1 Variance window. The results will be displayed in the Session window. In the test of Sigma-squared = vs > 0.02, is 53.70, and the p-value is Therefore, for any level of 0.017, the correct decision would be to not reject the null hypothesis.

20 Chapter 11 Chi-Square Tests 20 Suggested Exercises Section , 11.15, 11.17, 11.18, 11.19, Section , 11.29, 11.31, 11.32, 11.37, Section , 11.47, 11.48, Supplementary Exercises 11.50, 11.51, 11.53, 11.54, 11.58, 11.69, 11.72, 11.73, Technology Assignments TA 11.1, TA 11.2, TA 11.3