MG 620 Research and statistics for Managers Final Exam Instructions: Answer all questions Part I : Problem 1. Fitting a straight line to a set of data yields the following prediction line: Y = 10 + 4X i) Interpret the meaning of the Y intercept, b0. b0 = 10 is the value of Y when X is 0 ii) Interpret the meaning of the slope, b1 b1 = 4 mean that a unit change in X will increase Y by 4 units. iii) Predict the mean value of Y for X = 8 Y =10 +4X Y= 10 +5 (5) = 15 Dr. D Rawana Final Dec 2016 Page 1 2. What is the difference between the standard deviation, standard error of the mean, and standard error of the estimate? Discuss and offer examples. The standard deviation or SD describes the spread of values in the sample. The sample standard deviation s is a random quantity which varies from sample to sample. It stays the same on average when the sample size increases. The standard error of the mean or SE is the standard deviation of the sample mean and describes its accuracy as an estimate of the population mean. When the sample size increases, the estimator is based on more information and becomes more accurate, so its standard error decreases. 3. In a sample of size of 20, the sum of all values is 40. What is the sample mean a. 20 b. 40 SHOW YOUR WORK! c. d. 800 None of the above Mean =1/nXi Mean=1/20(40)=2 d Dr. D Rawana Final Dec 2016 Page 2 The following are the durations in minutes of a sample of long-distance phone calls within the continental United States reported by one long-distance carrier Table 1 Time (in minutes) Relative Frequency 0 but less than 5 5 but less than 10 10 but less than 15 15 but less than 20 20 but less than 25 25 but less than 30 30 or more 0.27 0.22 0.15 0.10 0.17 0.07 0.02 4. Referring to Table 1, what is the width of each class? a. 1 minute b. 2% SHOW YOUR WORK! c. d. 5 minutes 100% Interval width = Upper limit -Lower limit = 10-5 = 5minutes The correct choice is c 5. Referring to Table 1, what is the cumulative relative frequency for the percentage of calls that lasted under 10 minutes? a. 0.10 b. 0.76 SHOW YOUR WORK! c. d. 0.59 0.49 The term Cumulative Relative Frequency applies to an ordered set of observations from smallest to largest. It is the sum of the relative frequencies for all values that are less than or equal to the given value. .27 + .22 = .49 these two categories represent the relative frequencies that were under 10 mins. Dr. D Rawana Final Dec 2016 Page 3 6. The managers of a real estate firm are interested in finding out if the number of new clients a broker brings into the firm affects the sales generated by the broker. They sample 6 brokers and determine the number of new clients they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows. Broker Clients (X) 1 2 3 4 5 6 Sales (Y) 1 2 1 0 1 3 2 4 3 1 1 5 a. Construct a scatter diagram for these data. Does the scatter diagram show a linear relationship between sales and number of new clients? Explain and show all work including which variable is independent and dependent. Sales (Y) Y =1.437x + 0.75 R2 = 0.826 Y 6 5 4 3 2 1 0 1 2 3 X Clients (X) b) Estimate the intercept (b0). Show your work the coefficient of intercept is, Dr. D Rawana b0 0.75 Final Dec 2016 Page 4 c) Estimate the slope (b1). Show your work the coefficient of intercept is, b1 1.437 c) Draw the regression line y = 1.43 (x) + 0.75 d) Is the relationship positive or negative? Explain I believe that there is a weak positive relation occurring between the variables. 7. If SSR = 66 and SST = 88, and n = 10 a. Determine the coefficient of determination, r2, and interpret its meaning. r = SSR/SST 2 r =66/88=0.75 2 75 percent of the variation in the dependent variable can be explained in the independent variable. b. Compute the standard error of the estimate. SSE=SST-SSR=88-66-=22 Standard error= SSE = nk 22 =2.44 Standard error=2.44 101 a. How useful do you think this regression model is for predicting? I do believe it will be good for predicting. Dr. D Rawana Final Dec 2016 Page 5 8. The Management at Ohio National Bank does not want its customers to wait in line for service for too long. The manager of a branch of this bank estimated that the customers currently have to wait an average of 10 minutes for service. Assume that the waiting times for all customers at this branch have a normal distribution with a mean of 8 minutes and a standard deviation of 2 minutes. Find the probability that randomly selected customer will have to wait for less than 4 minutes? Instructions: Show all steps: 1. Draw the normal curve and indicate the mean, standard deviation, and the X bar scale 2. Identify the area of interest (that is shade the area under the curve that you will compute the probability). 3. Covert the X bar values in Z scores 4. Look up the Z standardized table for the cumulative area(s). 5. Now, make your decision ( that a customer will wait for less than 4 minutes) =8 dd Xbar 4 =10 410 ) 2 =P (Z<-3) Z<-3 1. P(X<4)=P(Z< P (Z<-3)= .0014 I believe that there is not a big chance that customers will be seen within 4 minutes. Dr. D Rawana Final Dec 2016 Page 6 9. The Management at Ohio National Bank does not want its customers to wait in line for service for too long. The manager of a branch of this bank estimated that the customers currently have to wait an average of 10 minutes for service. Assume that the waiting times for all customers at this branch have a normal distribution with a mean of 8 minutes and a standard deviation of 2 minutes. What is the probability that customers have to wait for 4 to 8 minutes? Instructions: Show all steps: a. Draw the normal curve and indicate the mean, standard deviation, and the X bar scale b. Identify the area of interest (that is shade the area under the curve that you will compute the probability). c. Covert the X bar values in Z scores d. Look up the Z standardized table for the cumulative area(s). e. Now, make your decision ( that a customer will have to wait for 4 to 8 minutes) xbar=4 =8 P (4X4)=P( 48 88 Z ) 2 2 =P (-2Z0) -2Z<0 P(-2Z<0)=0.5-0.0228=0.4772 I believe that with this probability result a customer may have to wait for 4 to 8 minutes 10. Five hundred employees were selected from a city's large private companies, and they were asked whether or not they have any pension benefits provided by their companies. Based on this information, the following two-way contingency table was prepared. Have Pension Benefits Yes Dr. D Rawana No Total Final Dec 2016 Page 7 Men 225 75 Women 150 50 Total If one employee is selected at random from these 500 employees, a) Find the probability that this employee is a man 150/300 = .3 b) Has pension benefits 75/500 = 0.15 c) Has pension benefits given the employee is a man 50/500 =0.1 d) Is a man given that he does have retirement benefits. 225/500 = 0.45 11. Use the given data to construct a frequency distribution. Lori asked 25 students how many hours they had spent doing homework during the previous week. The results are shown below: Number of students 1 2 3 Dr. D Rawana Hours of study 11 10 11 Final Dec 2016 Page 8 4 5 6 7 8 9 10 11 12 13 14 15 16 17 28 19 20 21 22 23 24 25 9 11 11 15 12 11 8 13 10 10 13 11 10 13 11 10 12 10 13 12 9 8 Construct a frequency distribution, using 4 classes and a class width of 2 hours, and a lower limit of 8 for class 1. There are 8 3 10 13 12 4 13 3 14 1 time Interval Frequency 8<10 3 0<12 13 12<14 7 Dr. D Rawana Final Dec 2016 Page 9 14<16 1 12. An entrepreneur is considering the sale of a Pizza-operated restaurant. The current owner claims that over the past 5 years, the average daily revenue was $650 with a standard deviation of $10. A sample of 36 days reveals daily revenue of $630. Consider H0: = 670 versus Ha: < 630 a. At the .05 level of significance , is there evidence that the average daily revenue is less than 630 b. Compute the p-value and interpret its meaning c. Construct a 95% confidence interval estimate of the population mean revenue of coin-operated laundry. d. Compare the results of (a) , (b), and (c). What conclusions do you reach? Show all steps! Instructions: Show all steps such as 1. State the Hypothesis 2. Draw the normal distribution and identify the acceptance and rejection regions 3. Make your decision with respect to each part of the question. 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