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Control Chart, Present Value, Linear Programming

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1. A state department of tourism and recreation collects data on the number of cars with out-of-state license plates in a state park. (The group's position is that more out-of-state plates means the state's advertising programs are working.) The sample size is fixed at n=100 each day. Data from the previous 20 days indicate the following number of out-of-state license plates:
Day Out-of-state Plates
1 24
2 33
3 18
4 21
5 26
6 31
7 15
8 18
9 23
10 13
11 22
12 15
13 25
14 33
15 16
16 22
17 34
18 10
19 23
20 13

a. Calculate the overall proportion of "tourists" (cars with out-of-state plates).
b. Calculate the LCL and UCL for these data.
c. Plot the control chart. Are all points within the control limits?
d. What action do you suggest for improving this process?

2. Cartons of Plaster of Paris are supposed to weigh exactly 32 oz. Inspectors want to develop process control charts. They take ten samples of six boxes and weigh them. Based on the following data, compute the lower and upper control limits and determine whether the process is in control.

Sample Mean Range
1 33.8 1.1
2 34.6 0.3
3 34.7 0.4
4 34.1 0.7
5 34.2 0.3
6 34.3 0.4
7 33.9 0.5
8 34.1 0.8
9 34.2 0.4
10 34.4 0.3

3. Larry's boat shop wants to monitor the number of blemishes in the paint of each boat. Construct a c-chart to determine if their paint process is in control using the following data.
Sample Number Number of
Defects
1 3
2 4
3 2
4 1
5 3
6 2
7 1
8 4
9 2
10 3

4. A firm is about to undertake the manufacture of a product, and is weighing the process configuration options. There are two different job shop processes under consideration, as well as a repetitive focus. The small job shop has fixed costs of $3,000 per month, and variable costs of $10 per unit. The larger job shop has fixed costs of $12,000 per month and variable costs of $2 per unit. A repetitive focus plant has fixed costs of $60,000 and variable costs of $0.50 per unit.
a. If demand were 1,000 units per month, what would cost be under each process configuration?
b. At what level of demand does the large job shop become cheaper than the small job shop?
c. At what level of demand does the repetitive shop become cheaper than the larger job shop?

5. The local convenience store makes personal pan pizzas. Currently, their oven can produce 50 pizzas per hour. It is has a fixed cost of $2,000, and a variable cost of $0.25 per pizza. The owner is considering a bigger oven that can make 75 pizzas per hour. It has a fixed cost of $3,000, but a variable cost of $0.20 per pizza.
a. What is the break-even point?
b. If the owner expects to sell 9,000 pizza, should he get the new oven?

6. Health Care Systems of the South is about to buy an expensive piece of diagnostic equipment. The company estimates that it will generate uniform revenues of $500,000 for each of the next eight years. What is the present value of this stream of earnings, at an interest rate of 6%? What is the present value if the machine lasts only six years, not eight? If the equipment cost $2,750,000, should the company purchase it?

7. A new machine tool is expected to generate receipts as follows: $5,000 in year one; $3,000 in year two, nothing in the next year, and $2,000 in the fourth year. At an interest rate of 6%, what is the present value of these receipts? Is this a better present value than $2,500 each year over four years? Explain.

8. A toy manufacturer has three different mechanisms that can be installed in a doll that it sells. The different mechanisms have three different setup costs (overheads) and variable costs, and therefore the profit from the dolls is dependent on the volume of sales. The anticipated payoffs are as follows.

Light Demand Moderate Demand Heavy Demand
Probability 0.25 0.45 0.3
Wind-up action $325,000 $190,000 $170,000
Pneumatic action $300,000 $420,000 $400,000
Electrical action ($400,000) $240,000 $800,000

a. What is the EMV of each decision alternative?
b. Which action should be selected?
c. What is the expected value under certainty?
d. What is the expected value of perfect information?

9. (A) The Queen City Nursery manufactures bags of potting soil from compost and topsoil. Each cubic foot of compost costs 12 cents and contains 4 pounds of sand, 3 pounds of clay, and 5 pounds of humus. Each cubic foot of topsoil costs 20 cents and contains 3 pounds of sand, 6 pounds of clay, and 12 pounds of humus. Each bag of potting soil must contain at least 12 pounds of sand, 12 pounds of clay, and 10 pounds of humus. Explain how this problem meets the conditions of a linear programming problem. Plot the constraints and identify the feasible region. Graphically or with corner points find the best combination of compost and topsoil that meets the stated conditions at the lowest cost per bag. Identify the lowest cost possible.

(B) A stereo mail order center has 8,000 cubic feet available for storage of its private label loudspeakers. The ZAR-3 speakers cost $295 each and require 4 cubic feet of space; the ZAR-2ax speakers cost $110 each and require 3 cubic feet of space; and the ZAR-4 model costs $58 and requires 1 cubic foot of space. The demand for the ZAR-3 is at most 20 units per month. The wholesaler has $100,000 to spend on loudspeakers this month. Each ZAR-3 contributes $105, each ZAR-2ax contributes $50, and each ZAR-4 contributes $28. The objective is to maximize total contribution. Write out the objective and the constraints. Do Not Solve.

10. Schriever Cabinet Specialties produces wall shelves, bookends, and shadow boxes. It is necessary to plan the production schedule for next week. The wall shelves, bookends, and shadow boxes are made of oak, of which the company currently has 600 board feet. A wall shelf requires 4 board feet, bookends require 2 board feet, and a shadow box requires 3 board feet. The company has a power saw for cutting the oak boards; a wall shelf requires 30 minutes, book ends require 15 minutes, and a shadow box requires 15 minutes. The power saw is available for 32 hours next week. After cutting, the pieces are hand finished in the finishing department, which consists of four skilled and experienced craftsmen, each of whom can complete any of the products. A wall shelf requires 30 minutes of finishing, bookends require 60 minutes, and a shadow box requires 90 minutes. The finishing department is expected to operate for 80 hours next week. Wall shelves sell for $29.95 and have a unit variable cost of $17.95; bookends sell for $11.95 and have a unit variable cost of $4.95; a shadow box sells for $16.95 and has a unit variable cost of $8.95. The company has a commitment to produce 10 wall shelf units for the Languages and Literature Department. The firm normally operates to achieve maximum contribution.

a. What are the decision variables of this problem (name them)?
b. What are the constraints of this problem (name them)?
c. Write out the objective and the constraints for this problem.
d. Solve the problem using Excel Solver

(please see attachment for complete set. thanks)

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Solution Summary

Answers 10 questions on Control Charts, p chart, X bar chart, R chart, c chart, Demand, Present Value, EMV, expected value under certainty, expected value of perfect information, Linear Programming.

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Testing Sleep Patterns

M = Mean
S2 = Population Variance
SM = Standard Deviation of the Distribution of Means
t = score for your sample
t needed = cut-off score that establishes the region of rejection (also known as the critical value)
Decision: Reject the Null or Fail to Reject the Null (select only one)

Evolutionary theories often emphasize that humans have adapted to their physical environment. One such theory hypothesizes that people should spontaneously follow a 24-hour cycle of sleeping and waking, even if they are not exposed to the usual pattern of sunlight. To test this notion, eight paid volunteers were placed (individually) in a room in which there was no light from the outside and no clocks or other indications of time. They could turn the lights on and off as they wished. After a month in the room, each individual tended to develop a steady cycle. Their cycles at the end of the study were as follows: 25, 27, 25, 23, 24, 25, 26, and 25.

Please provide answers indicated below

M =
S2 =
SM =
t =
t needed = +
Decision:

Below is an explanation of the symbols in Chapter 8, Practice Problem 18.

N1 = number of participants in the experimental group
N2 = number of participants in the control group
df1 = degrees of freedom for the experimental group
df2 = degrees of freedom for the control group
dfTotal = degrees of freedom for both groups
M1 = mean of the experimental group
M2 = mean of the control group
S21 = estimated population variance of the experimental group
S22 = estimated population variance of the control group
S2Pooled = pooled estimate of the population variance
S2M1 = variance of the distribution of means for the experimental group
S2M2 = variance of the distribution of means for the control group
S2Difference = variance of the distribution of differences between means
SDifference = standard deviation of the distribution of differences between means
t = score for your sample
t needed = cut-off score that establishes the region of rejection (also known as the critical value)
Decision: Reject the Null or Fail to Reject the Null (select only one)

18. Twenty students randomly assigned to an experimental group receive an instructional program; 30 in a control group do not. After 6 months, both groups are tested on their knowledge. The experimental group has a mean of 38 on the test (with an estimated population standard deviation of 3); the control group has a mean of 35 (with an estimated population standard deviation of 5).

Please provide answers indicated below

N1 =
N2 =
df1 =
df2 =
dfTotal =
M1 =
M2 =
S21 =
S22 =
S2Pooled =
S2M1 =
S2M2 =
S2Difference =
SDifference =
t =
t needed = +
Decision:

Chapter 9 Instructions
Practice Problem 17

Below is an explanation of the symbols in Chapter 9, Practice Problem 17.

S2Between = between groups population variance estimate
S2Within = within groups population variance estimate
F = statistical score that represents the ratio of the between groups to the within groups population variance estimate
F needed = cut-off score that establishes the region of rejection (also known as the critical value)
both groups
Decision: Reject the Null or Fail to Reject the Null (select only one)

17. Do students at various colleges differ in how sociable they are? Twenty-five students were randomly selected from each of three colleges in a particular region and were asked to report on the amount of time they spent socializing each day with other students. The results for College X was a mean of 5 hours and an estimated population variance of 2 hours; for College Y, M = 4, S2 = 1.5; and for College Z, M = 6, S2 = 2.5.

Please provide answers indicated below

S2Between =
S2Within =
F =
F needed = +
Decision:

Chapter 11 Instructions
Practice Problem 11 and 12c & 12d

For Chapter 11, Practice Problem 11, follow the instructions below to create 6 scatter plots using Microsoft Excel.

Note: The following instructions may not work with all versions of Excel. If you use Windows Vista, please scroll down to find instructions for creating scatter plots with the Windows Vista platform.

On most versions of Excel, do the following:

Open a blank Excel document. Click on Insert - then Chart. The chart Wizard will appear. Under the Standard Types tab, click on "XY (Scatter)"

Under the Chart Sub-Type: highlight the top box. It should be "Scatter, Compares pairs of values." click next, which should take you to "Chart Wizard - Step 2 of 4.

Click the Series tab. Click the Add button. In the Name Box, type the name of the scatter plot you will be designing, (i.e.) Perfect Positive, Perfect Negative, etc. In the X Values box, type in the values of the X coordinates you wish you use in your scatter plot. Separate numbers by commas. In the Y Values box, type in the values of the y coordinates you wish you use in your scatter plot. Separate numbers by commas. Click Next to go to Step 3 of 4. Do nothing in Step 3.

Click Next to go to Step 4 of 4 - Chart Location: In the Place Chart option, check "As New Sheet" Chart 1 should appear in the white box. Click OK, or Finish.

Click Insert to create the next chart - scatter plot. Your second scatter plot will be Chart 2; your third scatter plot will be Chart 3, etc.

Instructions for Windows Vista users are listed below:

On a new blank excel sheet, place your scores in two columns X and Y. Put these in rows A and B.
Click on the insert tab and in the charts sub-group, click the Scatter option. Click on the "scatter with only markers option."
This will present the blank chart and you are now in the Design mode. Second sub-group from the left is "Data." Click the "select data" button to open the 'select data source.'
Click hold and drag to select all the scores (this will be all the numbers in columns A and B). Click Ok and your scores will be put into the chart. Be sure not to select the "X" and the "Y".
Next to the "design" button is the "Layout" button.
In the layout mode, click on the "Axes" box and you can create the parameters of the horizontal and vertical axes. For example, lay your cursor over the Primary horizontal axes. At the bottom is "More primary horizontal axes options." Click on this to open the window to set your values, i.e. in axis options here you can set you minimum and maximum values as well as your units, e.g. .5 or 1. I found that your scores can be in decimal increments and will show just fine in a chart whose units are whole numbers.
Be sure and get the chart type you want, while in the "design" mode on the far left is "change chart type". Here you can be sure and select the scatter with markers only option. Also in the design mode, is chart layouts. Be sure you have selected the layout you want here as well.
To put chart title and such into your chart go to the lay out or format mode, the two options to the right of the design tab. On the far left is the current selection box. If you click on "chart area" the options for modifying specific descriptions within your chart such as the title, etc.
If you want to include a trendline, this option is in the "layout Mode," in the analysis button.
To change the "series 1" that appears in your chart to match your specific description, you have to go the 'design' mode and go to the 'select data' button. When the 'select data source' opens, click on 'series 1' to highlight and click 'edit' right above it. Here you can put your own info in and it will then appear on your chart. This will also create a chart title that will appear at the top of your chart.

Below is an explanation of the symbols in Chapter 11, Practice Problem 12c & 12d. Note: Instructions for Practice Problem 12c & 12d begin at the top of page 489 in your textbook (above problem 12)

r = score that represents the correlation coefficient
t = score that determines the significance of the correlation coefficient score
t needed = cut-off score that establishes the region of rejection (also known as the critical value)
both groups
Decision: Reject the Null or Fail to Reject the Null (select only one)

11. Make up a scatter diagram with 10 dots for each of the following situations: (a) perfect positive linear correlation, (b) large but not perfect positive linear correlation, (c) small positive linear correlation, (d) large but not perfect negative linear correlation, (e) no correlation, (f) clear curvilinear correlation.

12. Four research participants take a test of manual dexterity (high scores mean better dexterity) and an anxiety test (high scores mean more anxiety). The scores are as follows.
Person Dexterity Anxiety
1 1 10
2 1 8
3 2 4
4 4 -2

For problems 12 ( c) and 12 (d) do the following:
(c) figure the correlation coefficient;
(d) figure whether the correlation is statistically significant (use the .05 significance level, two-tailed)

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