Statistics Exam. July 2, Group... Bachelor in...

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1 Statistics Exam July 2, 2014 Name Number Group... Bachelor in A test to detect contamination in water is being developed. When a sample of contaminated water is tested, the test provides a positive result, indicating contamination, with probability 0.9. When a sample of non contaminated water is tested, the test provides a wrong result and indicates contamination with probability 0.1. If the probability that a sample of water is not contaminated is 0.95, what is the probability that the test provides a positive result (contamination) when the sample is really contaminated? 2. An exam consists of two questions. The first question is correctly answered by 60% of the students, while 50% of the students give the correct answer to the second. The students that answer correctly at least one of the two questions pass the exam. If 80% of the students pass the exam: a. What is the percentage of students that have answered both questions correctly? b. What is the probability that a student who answered correctly the first question does also give the correct answer to the second?

2 3. A device contains several components whose lifetimes in years (variable X) have density mass function: Answer the following questions: = for 0 a. What is the value of so that is actually a density mass function? b. What is the probability that a component lasts longer than 6 years? c. What is the probability that a component that has been working for 6 year lasts more than 6 more years?

3 d. The figure on the left shows four independent components that are assembled in parallel in a device. Compute the probability that the device lasts more than 6 years. Observation: The device works when at least one of the components work. 4. We are interested on the time spent until a call reaches its destination. Data on the time needed by each of 70 independent calls (in seconds) were provided by a Call Center. The result of a goodness-of-fit test is shown below: Histograma para Datos 15 Distribución Normal 12 frecuencia ,7 4,2 4,7 5,2 5,7 6,2 Datos Normal mean = 5,06495 standard deviation = 0, Normality Test Test Statistic p-value Chi-Square 17,6 0, Tail Areas Normal Distribution X Lower Tail Area (<) Upper Tail Area (>) 4,0 0, , ,5 0, , ,0 0, , ,5 0, , ,0 0, , Critical values Lower Tail Area Normal (<=) 0,01 3, ,1 4,4309 0,5 5, ,9 5, ,99 6,2159

4 Answer the following questions: a. Does the data fit a normal distribution?, why? b. What is the fitted normal distribution? c. What is the probability that a time between 4.5 and 5 seconds is needed until a call reaches its destination? d. The system fails if a call takes more than 6 seconds to reach its destination. What is the probability that no failure is detected in any of the next 10 calls? e. The system must be repaired if there is a failure in more than one of the next 10 calls. What is the probability that the system must be repaired?

5 5. A drug to reduce the level of cholesterol is tested on a group of 20 patients. The mean reduction of the cholesterol was 50 with a standard deviation of 15. The control group consists of 30 individuals who take a classical drug. The mean reduction on them is 40 with a standard deviation of 12. Is the new drug better than the classical one? p- value=0.012 (assuming equal variances). Justify your answer. Sample information H 0 H 1 Type of test p-value Conclusions 6. A production process is working satisfactorily if the percentage of defective produced items is at most 2%. A sample of size 100 is taken and it has 4 defective items. Is the production process working satisfactorily? p-value= Justify your answer. Sample information H 0 H 1 Type of test p-value Conclusions 7. In order control the weight of hamburgers, we have 20 trial samples of size 5 of hamburger weights (in grams). The data are below: Gráficos X-bar y R - Hamburgers Number of groups = 20 Group size = 5,0 Distribution: Normal Transformation: none R Chart Período #1-20 UCL: +3,0 sigma 19,7213 Center Line 9,32675 LCL: -3,0 sigma 0,0 1 out of limits X-bar Chart Período #1-20 UCL: +3,0 sigma 131,105 Center Line 125,725 LCL: -3,0 sigma 120,345 1 out of limits Estimations Período #1-20 Process mean 125,725 Process sigma 4,00978 Mean Range 9,32675

6 Gráfico X-bar para Hamburguesas Gráfico de Rangos para Hamburguesas , ,72 X-bar ,72 Rango , The statistics sample by sample are: 120, ,00 Group Size X-bar Range 1, ,959 7,531 2, ,449 8,138 3, ,035 6,135 4, ,727 4,48 5, ,834 6,756 6, ,188 8,839 7, ,186 12,764 8, ,78 8,279 9, ,983 8,282 10, ,644 * 21,189 11, ,258 4,687 12, ,246 9,869 13, ,853 7,674 14, ,497 11,145 15,0 5 * 132,4 13,52 16, ,049 7,374 17, ,443 10,564 18, ,443 7,199 19, ,766 12,295 20, ,759 9,815 For n=5, d 2 =2.326 ; D 3 =0 ; D 4 =2.115 For=4, d 2 =2.059 ; D 3 =0 ; D 4 =2.282 a. Build the control limits for the X-bar and Range charts.

7 b. If the specification limits are LSL=118 and USL=132, compute the process capacity (capability) and the process capability ratio (capacity index). c. If the process is going to be monitored by samples of size 4, compute the control limits of the X-bar and Range charts. 8. The charts and regressions at the end of the exam aim to explain the variable Resultados (Results, dependent variable) of several branches of a big firm in terms of their respective Sales (Ventas), Staff salaries (Personal), and Number of qualified workers (Cualificados). In the right column you can find the original data, while in the left column the data are transformed by taking logarithms. Answer the following questions: a. Should we transform the data? Answer Yes or No and continue solving the problem using the selected option. b. Write the simple linear regression equation of Resultados (results) in terms of Ventas (sales). Is the explanatory variable significant?, what is the impact of an increment of sales on the results?

8 c. Write and analyze the regression with the three explanatory variables. Is there any problem, why? d. What is the best regression model among those presented in the selected option (with or without logarithms). Analyze it and explain why you have chosen it.

9 Without logarithms With logarithms Resultados log(resultados) Ventas log(ventas) Personal log(personal) Cualificados log(cualificados) Regresión- Resultados Ventas CONSTANTE 36, ,0286 3, ,0004 Ventas 0, , , ,0000 R-cuadrada = 46,4149 porciento R-cuadrado (ajustado para g.l.) = 45,9187 porciento Regresión log(resultados) log(ventas) CONSTANTE 0, , , ,0493 log(ventas) 0, , , ,0000 R-cuadrada = 46,2571 porciento R-cuadrado (ajustado para g.l.) = 45,7595 porciento

10 Regresión - Resultados Personal CONSTANTE -178,985 22,7929-7, ,0000 Personal 1,7879 0, ,2939 0,0000 R-cuadrada = 62,069 porciento R-cuadrado (ajustado para g.l.) = 61,7178 porciento Regresión - Resultados Cualificados CONSTANTE -43, ,594-1, ,1220 Cualificados 3, , , ,0000 R-cuadrada = 24,9524 porciento R-cuadrado (ajustado para g.l.) = 24,2575 porciento Regresión - log(resultados) log(personal) CONSTANTE -7, , , ,0000 log(personal) 2, , ,6524 0,0000 R-cuadrada = 51,2357 porciento R-cuadrado (ajustado para g.l.) = 50,7842 porciento Regresión - log(resultados) log(cualificados) CONSTANTE -0, , , ,8618 log(cualificados) 1, , , ,0000 R-cuadrada = 20,6188 porciento R-cuadrado (ajustado para g.l.) = 19,883

11 Regresión Múltiple - Resultados Personal Cualificados CONSTANTE -172,906 23,2124-7, ,0000 Personal 1, , ,3907 0,0000 Cualificados -0, , , ,2018 R-cuadrada = 62,6448 porciento R-cuadrado (ajustado para g.l.) = 61,9466 porciento Regresión Múltiple - Resultados Personal Ventas CONSTANTE -142,997 22,4634-6,3658 0,0000 Personal 1, , , ,0000 Ventas 0, , , ,0000 R-cuadrada = 68,1158 porciento R-cuadrado (ajustado para g.l.) = 67,5198 porciento Regresión Múltiple - Resultados Cualificados Ventas CONSTANTE -35, ,1424-1, ,1145 Cualificados 1, , ,6059 0,0005 Ventas 0, , , ,0000 R-cuadrada = 52,2209 porciento R-cuadrado (ajustado para g.l.) = 51,3278 porciento Regresión Múltiple - log(resultados) log(cualificados) log(personal) CONSTANTE -7, , , ,0000 log(cualificados) -0, , , ,3883 log(personal) 2, , , ,0000 R-cuadrada = 51,5753 porciento R-cuadrado (ajustado para g.l.) = 50,6702 porciento Regresión Múltiple - log(resultados) log(personal) log(ventas) CONSTANTE -6, , , ,0000 log(personal) 1, , , ,0000 log(ventas) 0, , ,6715 0,0000 R-cuadrada = 62,5068 porciento R-cuadrado (ajustado para g.l.) = 61,806 porciento Regresión Múltiple - log(resultados) log(cualificados) log(ventas) CONSTANTE -1, , , ,0314 log(cualificados) 0, , ,8537 0,0002 log(ventas) 0, , , ,0000 R-cuadrada = 52,8072 porciento R-cuadrado (ajustado para g.l.) = 51,9251 porciento

12 Regresión Múltiple - Resultados Personal Cualificados Ventas CONSTANTE -138,203 22,7755-6, ,0000 Personal 1, , , ,0000 Cualificados -0, , , ,2351 Ventas 0, , , ,0000 R-cuadrada = 68,539 porciento R-cuadrado (ajustado para g.l.) = 67,6486 porciento Residuos de la regresión con tres variables independientes Regresión Múltiple - log(resultados) log(personal) log(ventas) log(cualificados) CONSTANTE -6, , , ,0000 log(personal) 1, , , ,0000 log(ventas) 0, , , ,0000 log(cualificados) -0, , , ,8220 R-cuadrada = 62,5248 porciento R-cuadrado (ajustado para g.l.) = 61,4642 porciento Residuos de la regresión con tres variables independientes. Gráfico de Residuos Gráfico de Residuos 5 4 Rediduo Estudentizado Rediduo Estudentizado predicho Resultados -4 2,3 3,3 4,3 5,3 6,3 predicho log(resultados)