Advanced Quantitative Methods - Essay Example

Seminar in Advanced Quantitative Methods I In each of questions through 3, a) identify the dependent and independent variables, b) the level of measurement for each variable, c) the value of the appropriate measure of association, and d) the proportionate reduction of error (PRE) interpretation of the association measure. 1. The following table considers the relationship between social class and health condition. Health Status Social Class Lower class Working class Middle or Upper Class Poor 9 17 11 Fair 11 58 37 Good or Excellent 14 133 150 A contingency table shows frequency distribution for bivariate data. There are two variables Health Status and Social Class in above contingency table. From above table, it appears that Health Status is dependent variable and Social Class is independent variable. The level of measurement for variable Health Status is ordinal and for variable Social Class is nominal. Table 1 shows the Chi-square Contingency Table Test for Independence using MegaStat’s (an Excel add-in). The Health Status depend on Social Class, 2(4, N = 440) = 25.61, p < .001. In other words, Health Status is dependent on Social Class. The appropriate measure of association for Health Status and Social Class is Cramers V coefficient. Cramers V = = = 0.1706 (n is sample size, and k is smaller of row or column) The value of Cramers V coefficient is 0.1706 that suggest there is weak relationship between Health Status and Social Class. The proportionate reduction of error (PRE), λ If we were asked to guess Heath Status, but we did not know the Social Status (independent variable) than the possible guess for Health Status is Good or Excellent because the mode for dependent variable Health Status is Good or Excellent. By doing this, we make 143 (= 440 - 297) mistakes out of the 440 choices, for an error rate of 143/440 or 32.5% (say old error). Now suppose we know both Health Status and Social Class (dependent and independent variable). Now the guess will that Lower Class has Good or Excellent Health Status, Working Class has Good or Excellent Health Status, and Middle or Upper Class has Good or Excellent Health Status. Now we make 143 (= (34 – 14) + (208 – 133) + (198 – 150)) mistakes out of the 440 choices, for an error rate of 143/440 or 32.5% (say new error). The difference between the old error and the new error, divided by the old error is the proportionate reduction of error, or PRE, λ . λ is not symmetric its value depends on which is the independent variable. The proportionate reduction of error (PRE), λ suggests that the Health Status and Social Class are independent. Table 1 Chi-square Contingency Table Test for Independence Lower class Working class Middle or Upper Class Total Poor Observed 9 17 11 37 Expected 2.86 17.49 16.65 37.00 (O - E)² / E 13.19 0.01 1.92 15.12 Fair Observed 11 58 37 106 Expected 8.19 50.11 47.70 106.00 (O - E)² / E 0.96 1.24 2.40 4.61 Good or Excellent Observed 14 133 150 297 Expected 22.95 140.40 133.65 297.00 (O - E)² / E 3.49 0.39 2.00 5.88 Total Observed 34 208 198 440 Expected 34.00 208.00 198.00 440.00 (O - E)² / E 17.64 1.65 6.32 25.61 25.6075 chi-square 4 df 0.0000 p-value 0.2412 Phi coefficient 0.1706 Cramérs V 2. The use of cellular phones in automobiles has increased dramatically in the last few years. Of concern to traffic experts, as well as manufacturers of cellular phones, is the effect on accident rates. Is someone who is using a cellular phone more likely to be involved in a traffic accident? What is your conclusion from the following sample information? Use the .05 significance level.   Had Accident in the Last Year Did Not Have an Accident in the Last Year Cellular phone in use 35 200 Cellular phone not in use 50 400 Use of cellular phone is independent variable and traffic accident is dependent variable. The level of measurement for both variables use of cellular phone and traffic accident is nominal. Table 2 shows the Chi-square Contingency Table Test for Independence using MegaStat’s (an Excel add-in). The traffic accident is independent of use of cellular phone, 2(1, N = 600) = 2.03, p = .154. In other words, someone who is using a cellular phone is not more likely to be involved in a traffic accident. The appropriate measure of association for use of cellular phone and traffic accident is Phi (φ) coefficient. The value of Phi (φ) coefficient is 0.0545 that suggest there is no association between use of cellular phone and traffic accident. The proportionate reduction of error (PRE), λ Here, column represents the dependent variable (traffic accident). Therefore, if no information is available for cellular phone use than error is Misclassified in situation 1 (E1) = 85 If information is available for cellular phone use than error is Misclassified in situation 2 (E2) = (235 – 200) + (450 – 400) = 85 Therefore, the proportionate reduction of error, or PRE, λ is The proportionate reduction of error (PRE), λ suggests that use of cellular phone and traffic accident are independent. Table 2 Chi-square Contingency Table Test for Independence Had Accident in the Last Year Did Not Have an Accident in the Last Year Cellular phone in use Observed 35 200 Expected 29.16 205.84 (O - E)² / E 1.17 0.17 Cellular phone not in use Observed 50 400 Expected 55.84 394.16 (O - E)² / E 0.61 0.09 Total Observed 85 600 Expected 85.00 600.00 (O - E)² / E 1.78 0.25 2.0322 chi-square 1 df .1540 p-value .0545 Phi coefficient .0545 Cramérs V 3. A researcher believes that women are happier in marriage than men. To examine the relationship between sex and marital happiness, the following table was obtained for a sample of 300 adults. Happiness Rating Men Women Very Happy 103 75 Pretty Happy 51 60 Not Too Happy 5 6 Sex (men or women) is independent variable and Marital Happiness Rating is dependent variable. The level of measurement for variable Marital Happiness Rating is ordinal and for variable Sex (men or women) in nominal. Table 3 shows the Chi-square Contingency Table Test for Independence using MegaStat’s (an Excel add-in). Marital happiness does not differ by sex, 2(2, N = 300) = 4.16, p = .125. In other words, the men and women have same marital happiness. The appropriate measure of association for Sex and Marital Happiness Rating is Cramers V coefficient. Cramers V = = = 0.1178 (n is sample size, and k is smaller of row or column) The value of Cramers V coefficient is 0.1178 that suggest there is very weak (not useful) relationship between Sex and Marital Happiness Rating. The proportionate reduction of error (PRE), λ Here, row represents the dependent variable (Happiness Rating). Therefore, if no information is available about sex than error is Misclassified in situation 1 (E1) = 300 – 178 = 122 If information is available about sex than error is Misclassified in situation 2 (E2) = (159 – 103) + (141 – 75) = 122 Therefore, the proportionate reduction of error, or PRE, λ is The proportionate reduction of error (PRE), λ suggests that sex and marital happiness are independent. Table 3 Chi-square Contingency Table Test for Independence Happiness Rating Men Women Total Very Happy Observed 103 75 178 Expected 94.34 83.66 178.00 (O - E)² / E 0.79 0.90 1.69 Pretty Happy Observed 51 60 111 Expected 58.83 52.17 111.00 (O - E)² / E 1.04 1.18 2.22 Not Too Happy Observed 5 6 11 Expected 5.83 5.17 11.00 (O - E)² / E 0.12 0.13 0.25 Total Observed 159 141 300 Expected 159.00 141.00 300.00 (O - E)² / E 1.96 2.20 4.16 4.1601 chi-square 2 df .1249 p-value 0.1178 Phi coefficient 0.1178 Cramérs V 4. A group of citizens has filed a complaint with the Chief of the Houston Police Department alleging that poor neighborhoods receive significantly less protection than more affluent neighborhoods. The Chief asks you to examine the complaint using the percent of the population on welfare and the number of hours of daily police patrol in a sample of communities in Houston. Community Number Percent on Welfare Hours of Daily Police Patrol 1 40 20 2 37 15 3 32 20 4 29 20 5 25 15 6 24 20 7 17 15 8 15 20 9 2 50 10 4 40 a. What are the dependent and independent variables for this study? The dependent and independent variables for the study are Hours of Daily Police Patrol and Percent of the Population on Welfare, respectively. b. Use EXCEL or SPPS to produce a regression analysis. Regression Statistics Multiple R 0.7589 R Square 0.5760 Adjusted R Square 0.5230 Standard Error 8.1478 Observations 10 ANOVA   df SS MS F Significance F Regression 1 721.4114 721.4114 10.8669 0.0109 Residual 8 531.0886 66.3861 Total 9 1252.5000   Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 39.0700 5.3803 7.2617 0.0001 26.6631 51.4769 Percent on Welfare -0.6920 0.2099 -3.2965 0.0109 -1.1761 -0.2079 c. Find, and interpret, the value of the slope in the context of this study. The value of the slope is -0.692. The value of slope suggests that every percent increase in percent of the population on welfare drops about 0.7 hours of daily police patrolling. d. Find, and interpret, the value of the y-intercept in the context of this study. The value of the y-intercept is 39.07. The value of y-intercept suggests that population without welfare has about 39 hours of daily police patrolling. e. What is the prediction (also called least squares) equation? The prediction equation is Hour of Daily Police Patrol = 39.07 – 0.692(Percent on Welfare) f. What is the expected number of hours of daily police patrol for a community in which 30 percent is on welfare. Hour of Daily Police Patrol = 39.07 – 0.692(30) = 18.31 The expected number of hours of daily police patrol for a community in which 30 percent is on welfare is about 18 hours. 5. 5. As a researcher, you want to investigate the hypothesis that the number of crimes is related to police expenditures per capita because states with higher crime rates are likely to increase their police force, thereby spending more on the number of officers on the street. The following table shows X is the number of crimes (in thousands) and Y is Police Protection Expenditures per Capita (Dollars), per 10,000 people in the eastern and Midwestern United States. State X = number of crimes Y = Police Protection Expenditures per Capita Maine 2.88 122.50 New Hampshire 2.28 141.80 Vermont 2.81 102.80 Massachusetts 3.26 218.70 Rhode Island 3.58 179.20 Connecticut 3.38 193.60 New York 3.27 292.40 New Jersey 3.40 236.60 Pennsylvania 3.11 171.20 Ohio 4.00 179.40 Indiana 3.76 124.50 Illinois 4.51 224.40 Michigan 4.32 172.30 Wisconsin 3.29 196.60 Minnesota 3.59 166.80 Iowa 3.22 135.80 Missouri 4.57 153.90 North Dakota 2.39 102.90 South Dakota 2.64 115.30 TOTAL 64.26 3,230.70 a) Find the least-squares regression equation that predicts police expenditures per capita from the number of crimes, and hence predict the police expenditure per capita for a state with 3.65 crimes (in thousands). The least-squares regression equation that predicts police expenditures per capita from the number of crimes is Police Protection Expenditures per Capita = 73.06 + 28.67(Number of Crimes) Where, the number of crimes is in thousands and Police Protection Expenditures per Capita (Dollars) is per 10,000 people. Police Protection Expenditures per Capita = 73.06 + 28.67(3.65) = 177.72 The police expenditure per capita for a state with 3.65 crimes (in thousands) is S177.72 per 10,000 people. b) Find, and interpret (in terms of police expenditures per capita and the number of crimes), the coefficient of determination. The coefficient of determination is 0.1402. This value suggests that the number of crimes explains about 14% variation in police expenditure per capita for a state. The number of crimes does not explain the other 86% variation in police expenditure per capita for a state. Regression Statistics Multiple R 0.3745 R Square 0.1402 Adjusted R Square 0.0897 Standard Error 47.5803 Observations 19 ANOVA   df SS MS F Significance F Regression 1 6277.7787 6277.7787 2.7730 0.1142 Residual 17 38485.9855 2263.8815 Total 18 44763.7642         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 73.0594 59.2506 1.2331 0.2343 -51.9485 198.0672 number of crimes 28.6737 17.2190 1.6652 0.1142 -7.6552 65.0026 Read More