An analyst has drawn twelve samples, each containing six sample points, from a process that produces a product. The diameter of each product in the samples was determined. The results were tabulated and sample means and ranges were computed. The results were:

Sample

Sample Mean (in.) Range (in.)

1 10.012 0.009

2 10.007 0.014

3 9.991 0.007

4 10.006 0.026

5 9.997 0.016

6 9.999 0.012

7 10.001 0.008

8 10.005 0.013

9 9.995 0.015

10 10.001 0.011

11 10.003 0.014

12 10.006 0.009

a) Determine the upper and lower control limits and the overall means for x-charts and R-charts. b) Draw the charts and plot the values of the sample means and ranges. c) Do the data indicate a process that is in control? d) Why or why not?

Question 6.2 A manufacturer produces three types of humidifiers. The retail price of Humidifier Regular (HR) is $60 with variable costs of $20. Humidifier Advanced (HA) sells for $200 for its advanced features. This models variable costs are $80. Humidifier Simple (HS) is a simplified version and only sells for $25 with variable costs of $15. The manufacturer has machines and facilities worth $320,000 annually. The sales data show that 1000 units of HR, 2000 units of HA, and 10,000 units of HS was sold last year. Calculate the break-even point of the firm. The firm has some idle capacity at these volumes, and chooses to cut the selling price of HR from $60 to $45, believing that its sales volume will rise from 1000 units to 2500 units. What is the revised break-even point?

Question 6.3 Leonards company purchases a window hardware from Williams Hardware Co and sells to the market. Leonard is also considering manufacturing the product internally. Williams Hardware Co. charges $4 per unit, with a minimum order of 3,000 units. If Leonard manufactures the product in-home, the process setup fee could be $15,000 and then $1.82 per unit for labor and materials. a) Draw a graph illustrating the crossover (or indifference) point.