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 model's 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

Leonard's company purchases a window hardware from William's Hardware Co and sells to the market. Leonard is also considering manufacturing the product internally.William's 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.

b) Determine the number of units where either choice has the same cost.