Find UCL and LCL. Is the number of errors in a state of statistical control?

Transcription

1 1) A private mail delivery service has a policy of guaranteeing delivery by 10:30 A.M. of the morning after a package is picked up. Suppose that management of the service wishes to study delivery performance in a particular geographic area over a 3-week time period based on a 5-day work week. The total number of packages delivered daily an d the number of packages that were not delivered by 10:30 A.M. are recorded and the results are as follows: Day Packages Delivered Packages not Arriving Proportion before 10:30 AM a) Determine the upper and lower control limits. b) Does the process given an out-of-control signal? 2) The branch manager of a saving bank has recorded the number of errors of a particular type that each of 16 tellers has made during the past year. The results are as follows: Teller Teller s Number of Teller Teller s Number of Number Errors Number Errors Alice 1 4 Olivia 9 6 Carl 2 7 Pat 10 2 Gina 3 15 Paul 11 5 Jane 4 6 Susan 12 3 Linda 5 2 Thomas 13 7 Marla 6 5 Vera 14 5 Mitchell 7 6 Victoria Nora 8 7 Walter 16 4 Find UCL and LCL. Is the number of errors in a state of statistical control? 1

2 3) Given the following summary data for subgroups of n = 4 for a 10-day period. Day Average Range Day Average Range a) Calculate the upper and lower control limits for the range chart. Is the range in control? b) Calculate the upper and lower control limits for the mean (x) chart. Are means under control? 4) Sales of Humber Co. over the past year are as follows (in $1000): a) Develop a 3-month moving average. April May June July August September October November December January February March b) Using a), determine the demand for April c) Develop an exponential smoothing with α = 0.6 d) Using c), develop the demand for April

3 5) An analyst for energy Consumption Survey is trying to develop a model to predict the consumption of heating oil during a month is East Finch. He constructed the multiple regression equation of home heating oil, y, in terms of average daily atmospheric temperature outside the house, x 1, and amount of insulation in the attic of the house, x 2, during a month. The regression equation for a random sample is 15 homes is as follows: y = x x 2 where y is measured in gallons, x 1 in Fahrenheit degrees, F, and x 2 in inches. The total variation SST is equals to , and unexplained variation SSE is equal to a) Calculate the coefficient of determination in R 2 and explain its meaning. b) Calculate the adjusted coefficient of determination (adjusted R 2 ). c) Explain the meaning of the coefficient d) Predict the number of gallons of heating consumed in a house that had 6 inches of attic insulation during November in which the average daily atmospheric temperature was 30 F. 6) Does the process give an out-of-control signal? Why? 3

4 7) Does the process give an out-of-control signal? Why? 4

5 Definitions and concepts to know 1. Time series 2. Trend component 3. Cyclical component 4. Seasonal component 5. Random variation component 6. Moving averages 7. Weighted moving averages 8. Exponential smoothing, smoothing constant 9. R-chart 10. x-chart 11. p-chart 12. c-chart 13. Assignable variation point Rule Rule and control limits 16. Correlation 17. Simple and multiple regression 18. Coefficient of determination 19. Adjusted coefficient of determination 20. Explained variation (SSR), unexplained variation (SSE), and total variation (SST). 5

6 Answers/Solutions 1) a) UCL = 0.227, LCL = b) The process is out of statistical control. The day 2 is below the lower limit, and the day 10 is above the upper control limit. 2) UCL = , LCL does not exist The number of errors is not in a state of statistical control. Gina made 15 errors. 3) a) UCL = ; Range is in statistical control. 4) a) b) UCL = , LCL = The process mean is out of statistical control. The day 7 and 5 are outside the control limits. Period Sales Moving Averages April May June July August September October November December January February March b) Forecast for April 2006 is 37 c) Period Sales Exponential Smoothing April May June July August September October November December January February March

7 d) Forecast for April 2006 = ) a) R 2 = The value R 2 = indicates that 96.56% of the variation in consumption of home heating oil can be explained by moving average daily atmospheric temperature outside the house x 1 and amount of insulation in the attic of the house x 2 during a month. b) adjusted R 2 = c) The slope of amount of attic insulation b 2 means that for a month with a given average daily atmospheric temperature, the consumption of heating oil will decrease by gallons for each additional inch of attic insulation. d) gallons 6) The following control chart shows that the process is out of statistical control because values beyond the control limits. 7) The following control chart shows that the process is out of statistical control because there are 8 consecutive points all above (below) the center line. 7