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STA 213, Fall 2011, StatCrunch Assignment #5 - Solution Exercise 13.90 π1: true proportion of all customers that purchased the warranty when the product was sold at the regular price π2: true proportion of all customers that purchased the warranty when the product was sold at the sale price Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .10 ( p p ˆ 1 ˆ2) Calculations: Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 47 229 25 178 0.06479073 0.03813 1.6992 0.0446 [So, accept HA ] Interpretation: At the 10% significance level there is sufficient evidence to say the true proportion of all customers purchasing the warranty is higher when the product is sold at the regular price than at the sale price. Exercise 13.92 p1: true proportion of all people with credit score below 600 that default on their loan p2: true proportion of all people with credit score 600 and above that default on their loan. Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .10 ( p p ˆ 1 ˆ2) Calculations: Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 11 562 7 804 0.0109 0.00627 1.733 0.0415 [So, accept HA ] Interpretation: At the 10% significance level there is sufficient evidence to say the true proportion of borrowers with credit score below 600 that default on their loan is greater than the true proportion of all borrowers with credit score 600 and above that default on their loan. Yes, people with lower credit score are more likely to default. Exercise 13.101 p1: true proportion of all Lexus owners that plan to buy another Lexus for their next car. p2: true proportion of all Acura owners that plan to buy another Acura for their next car. Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 ≠ 0; ˆ ˆ Accept HA if P-value < = .05 ( p p ˆ 1 ˆ2) Calculations: Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 317 350 261 294 0.01796 0.02399 0.7485 0.4541 [So, do not accept HA ] Interpretation: At the 5% significance level there is not sufficient evidence to say the true proportion of all Lexus owners that would buy another Lexus is any different from the true proportion of all Acura owners that would buy another Acura. Thus, we cannot say they have different satisfaction levels. Exercise 13.102 a. p1: true proportion of all smokers that suffer from some form of heart disease; p2: true proportion of all non-smokers that suffer from some form of heart disease; Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .10 ( p p ˆ ˆ 1 2) Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 10 38 12 162 0.18908381 0.056397 3.35273 0.0004 [So, accept HA ] Interpretation: At the 10% significance level there is sufficient evidence to say the true proportion of all smokers that suffer some form of heart disease is greater than the true proportion of non-smokers that suffer some form of heart disease. b. p1: true proportion of all smokers that suffer from some form of heart disease; p2: true proportion of all non-smokers that suffer from some form of heart disease; Formula: ( p1 p2 ) Z ( p1 p2 ) ˆ ˆ ˆ ˆ 90% confidence interval results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. L. Limit U. Limit p1 - p2 10 38 12 162 0.18908381 0.074338 0.066808 0.31136 Interpretation: We estimate with 90% confidence the true proportion of all smokers that suffer some form of heart disease is between 6.68% and 31.4% greater than the true proportion of all non-smokers that suffer some form of heart disease. Exercise 13.106 p1: true proportion of all workers with just a high school education that work 11 or more hours per day p2: true proportion of all workers with a post secondary education that work 11 or more hours per day Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .05 ( p p ˆ ˆ 1 2) Calculations: Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 167 194 63 80 0.07332474 0.048783 1.5030777 0.0664 [So, do not accept HA ] Interpretation: At the 5% significance level there is not sufficient evidence to say the true proportion of all workers with just a high school education that work 11 or more hours per day is any greater than the proportion of all workers with a post secondary education that work 11 or more hours per day. We cannot infer that those with more education are less likely to work 11 or more hours per day. Exercise 13.126 a. p1: true proportion of all boys in 2001 that participated in organized sports p2: true proportion of all boys in 2011 that participated in organized sports Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .05 ( p p ˆ ˆ 1 2) Calculations: Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 178 271 174 313 0.10091603 0.04060 2.4854672 0.0065 [So, do accept HA ] Interpretation: At the 5% significance level there is sufficient evidence to say the true proportion of all boys in 2001 that participated in organized sports is greater than the true proportion of all boys in 2011 that participated in organized sports. We can infer that sports participation has decreased for boys in the last decade. b. p1: true proportion of all girls in 2001 that participated in organized sports p2: true proportion of all girls in 2011 that participated in organized sports Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .05 ( p p ˆ ˆ 1 2) Calculations: Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 137 281 137 304 0.036886588 0.04129 0.8932662 0.1859 [So, do not accept HA ] Interpretation: At the 5% significance level there is not sufficient evidence to say the true proportion of all girls in 2001 that participated in organized sports is greater than the true proportion of all girls in 2011 that participated in organized sports. We cannot infer that sports participation has decreased for girls in the last decade. c. p1: true proportion of all boys in 2011 that participated in organized sports p2: true proportion of all girls in 2011 that participated in organized sports Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .05 ( p p ˆ ˆ 1 2) Calculations: Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 174 313 137 304 0.105252646 0.040261 2.6142292 0.0045 [So, do accept HA ] Interpretation: At the 5% significance level there is sufficient evidence to say the true proportion of all boys in 2011 that participated in organized sports is greater than the true proportion of all girls in 2011 that participated in organized sports. Yes, we can infer that girls are less likely to participate in organized sports this year. Exercise 13.132 p1: true proportion of all drivers that wear their seat belts last year p2: true proportion of all drivers that wear their seat belts this year Test Statistic: Z ( p1 p2 ) 0 ; H0 : p1 - p2 = 0; HA : p1 - p2 > 0; ˆ ˆ Accept HA if P-value < = .10 ( p p ˆ ˆ1 2) Hypothesis test results: Difference Count1 Total1 Count2 Total2 Sample Diff. Std. Err. Z-Stat P-value p1 - p2 221 327 288 382 -0.07808572 0.03390 -2.30317 0.0106 [So, accept HA ] Interpretation: At the 5% significance level there is sufficient evidence to say the true proportion of all drivers that wear their seat belts last year is less than the true proportion of all drivers that wear their seat belts this year. Thus, yes there is evidence of significantly higher seat belt usage.

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