Identify the level of measurement of each of the following variables (Nominal, Ordinal, or Scale (Ratio):

1.  County names in a state

2. Number of participants in food stamp programs.

3. Reputations of colleagues ranked on the scale of Very High to Very Low.

4. Leadership ability measured on a scale from 0 to 5.

5. Divisions within a state agency.

6. Inventory broken down into three categories: tightly controlled, moderately controlled, or minimally controlled.

About 40% of all US adults will try to pad their insurance claims. Suppose that you are the director of an insurance adjustment office. Your office has just received 128 insurance claims to be processed int the next few days. What is the probability that -

half or more claims have been padded?

fewer than 45 of the claims have been padded:

From 40 to 64 of the claims have been padded?

More than 80 of the claims have not been padded?

The consumer food database contains five variables: Annual Food Spending per Household, Annual Household Income, Non-Mortgage Household Debt, Geographic Region of the U.S. of the Household, and Household Location. There are 200 entries for each variable in this database representing 200 different households from various regions and locations in the United States. Annual Food Spending per Household, Annual Household Income, and Non-Mortgage Household Debt are all given in dollars. The variable Region tells in which one of four regions the household resides. In this variable, the Northeast is coded as 1, the Midwest is coded 2, the South is coded as 3, and the West is coded as 4. The variable Location is coded as 1 if the household is in a metropolitan area and 2 if the household is outside a metro area. The data in this database were randomly derived and developed based on actual national norms.The consumer food database contains five variables: Annual Food Spending per Household, Annual Household Income, Non-Mortgage Household Debt, Geographic Region of the U.S. of the Household, and Household Location. There are 200 entries for each variable in this database representing 200 different households from various regions and locations in the United States. Annual Food Spending per Household, Annual Household Income, and Non-Mortgage Household Debt are all given in dollars. The variable Region tells in which one of four regions the household resides. In this variable, the Northeast is coded as 1, the Midwest is coded 2, the South is coded as 3, and the West is coded as 4. The variable Location is coded as 1 if the household is in a metropolitan area and 2 if the household is outside a metro area. The data in this database were randomly derived and developed based on actual national norms.

Provide a 1,600-word detailed, statistical report including the following:

• Explain the context of the case
• Provide a research foundation for the topic
• Present graphs
• Explain outliers
• Prepare calculations
• Conduct hypotheses tests
• Discuss inferences you have made from the results

This assignment is broken down into four parts:

• Part 1 - Preliminary Analysis
• Part 2 - Examination of Descriptive Statistics
• Part 3 - Examination of Inferential Statistics
• Part 4 - Conclusion/Recommendations

Part 1 - Preliminary Analysis (3-4 paragraphs)

Generally, as a statistics consultant, you will be given a problem and data. At times, you may have to gather additional data. For this assignment, assume all the data is already gathered for you.

State the objective:

• What are the questions you are trying to address?

Describe the population in the study clearly and in sufficient detail:

• What is the sample?

Discuss the types of data and variables:

• Are the data quantitative or qualitative?
• What are levels of measurement for the data?

Part 2 - Descriptive Statistics (3-4 paragraphs)

Examine the given data.

Present the descriptive statistics (mean, median, mode, range, standard deviation, variance, CV, and five-number summary).

Identify any outliers in the data.

Present any graphs or charts you think are appropriate for the data.

Note: Ideally, we want to assess the conditions of normality too. However, for the purpose of this exercise, assume data is drawn from normal populations.

Part 3 - Inferential Statistics (2-3 paragraphs)

Use the Part 3: Inferential Statistics document.

• Create (formulate) hypotheses
• Run formal hypothesis tests
• Make decisions. Your decisions should be stated in non-technical terms.

Hint: A final conclusion saying "reject the null hypothesis" by itself without explanation is basically worthless to those who hired you. Similarly, stating the conclusion is false or rejected is not sufficient.

Part 4 - Conclusion and Recommendations (1-2 paragraphs)

Include the following:

• What are your conclusions?
• What do you infer from the statistical analysis?
• State the interpretations in non-technical terms. What information might lead to a different conclusion?
• Are there any variables missing?
• What additional information would be valuable to help draw a more certain conclusion?

A group of high-school parents in Tucson, Arizona, in conjunction with faculty from the University of Arizona, claim that young women in the Tucson high schools not only are called on less frequently, but receive less time to interact with the instructor than do young men. They would like to see the school district hire a coordinator, spend money (and time) on faculty workshops, and offer young women classes on assertiveness and academic communication.

To make things simple, assume that instructor interactions with young men average 95 seconds, with standard deviation 35 seconds. (Treat this as population information.)

The null hypothesis will be that the average interaction time for young women will also be 95 seconds, as opposed to the alternate hypothesis that it is less, and will be tested at the 2.5% level of significance.

1. Give interpretations in context of Type I and Type II error in this situation. (Your discussion should not focus on “Null” and “Alternate”.)
2. What are the social, economic, and other consequences of (separately) Type I and Type II error?
3. Find the rejection region for this test. That is, what interaction time bounds the lower 2.5% of the distribution?
4. Assume the true mean interaction time for young women is 90 seconds. Find the power of the test.
5. Repeat part 4 for a true mean interaction time of 80 seconds.
6. What do the results in parts (4) and (5) mean in terms of your previous answers?

Indicate whether each statement represents a conceptual definition, part of an operational definition, or a hypothesis.

1. The organizational capacity of nonprofit organizations consists of their relevance, responsiveness, effectiveness, and resilience.

2. To determine the quality of police services, we asked respondents if they thought there was less, about the same, or more crime in their neighborhoods compared to the rest of the city.

3. The more politically engaged the respondents, the higher the probability that they will have a favorable attitude toward government services.

4. Home health care has been identified as an array of therapeutic and preventive services provided to patients in their homes or in foster homes.

5. Controlled items are those that must be identified, accounted for, secured, segregated, and handled in a special manner.

6. Uncontrolled items are likely to have a higher rate of wastage than controlled items.

1.

Given the following contingency table, conduct a test for independence at the 1% significance level. (You may find it useful to reference the appropriate table: chi-square table or F table)

Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

2.

A market researcher for an automobile company suspects differences in preferred color between male and female buyers. Advertisements targeted to different groups should take such differences into account if they exist. The researcher examines the most recent sales information of a particular car that comes in three colors. (You may find it useful to reference the appropriate table: chi-square table or F table)

Calculate the value of the test statistic. (Round the intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

3.

Consider the following sample data with mean and standard deviation of 20.1 and 7.3, respectively. (You may find it useful to reference the appropriate table: chi-square table or F table)

Calculate the value of the test statistic. (Round the z value to 2 decimal places, all other intermediate values to at least 4 decimal places and final answer to 3 decimal places.)

The average number of cavities that thirty-year-old Americans have had in their lifetimes is 5. Do twenty-year-olds have more cavities? The data show the results of a survey of 13 twenty-year-olds who were asked how many cavities they have had. Assume that the distribution of the population is normal.

4, 7, 4, 6, 5, 4, 5, 5, 5, 5, 5, 4, 4

What can be concluded at the αα = 0.01 level of significance?

1. For this study, we should use Select an answer t-test for a population mean z-test for a population proportion
2. The null and alternative hypotheses would be:

H0:H0:  ? μ p  ? = ≠ < >       

H1:H1:  ? p μ  ? = ≠ < >    

3. The test statistic ? z t  =  (please show your answer to 3 decimal places.)
4. The p-value =  (Please show your answer to 4 decimal places.)
5. The p-value is ? ≤ >  αα
6. Based on this, we should Select an answer fail to reject reject accept  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest that the population mean number of cavities for twenty-year-olds is not significantly more than 5 at αα = 0.01, so there is insufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is more than 5.
   • The data suggest the population mean is not significantly more than 5 at αα = 0.01, so there is sufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is equal to 5.
   • The data suggest the populaton mean is significantly more than 5 at αα = 0.01, so there is sufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is more than 5.
8. Interpret the p-value in the context of the study.
   • There is a 72.5686636% chance of a Type I error.
   • If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds then there would be a 72.5686636% chance that the population mean number of cavities for twenty-year-olds would be greater than 5.
   • If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds then there would be a 72.5686636% chance that the sample mean for these 13 twenty-year-olds would be greater than 4.85.
   • There is a 72.5686636% chance that the population mean number of cavities for twenty-year-olds is greater than 5.
9. Interpret the level of significance in the context of the study.
   • If the population mean number of cavities for twenty-year-olds is more than 5 and if you survey another 13 twenty-year-olds, then there would be a 1% chance that we would end up falsely concuding that the population mean number of cavities for twenty-year-olds is equal to 5.
   • If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds, then there would be a 1% chance that we would end up falsely concuding that the population mean number of cavities for twenty-year-olds is more than 5.
   • There is a 1% chance that flossing will take care of the problem, so this study is not necessary.
   • There is a 1% chance that the population mean number of cavities for twenty-year-olds is more than 5.

Debate if “failing to reject the null” is the same as “accepting the null.” Support your position with examples of acceptance or rejection of the null.

Generate a simulated data set with 100 observations based on the following model. Each data point is a vector Z= (X, Y) where X describes the age of a machine New, FiveYearsOld, and TenYearsOld and Y describes whether the quality of output from the machine Normal or Abnormal. The probabilities of a machine being in the three states are

P(X = New) = 1/4

P(X = FiveYearsOld) = 1/3

P(X = TenYearsOld) = 5/12

The probabilities of Normal output conditioned are machine age are

P(Y = Normal | X= New) = 8/10

P(Y = Normal | X= FiveYearsOld) = 8/10

P(Y = Normal | X= TenYearsOld) = 4/10

Your data should consist of two vectors Y and Z both of which are of class character. Convert these to factors using the as.factor function. Analyze your simulated data using the chisq.test function with inputs x=x, y=y. Perform the analysis with the exact same function, but with simulated p-values using the inputs x=x, y=y, simulate.p.values=TRUE, B=10000. Would you trust the p-values from the asymptotic distribution or the simulated p-values more? What conclusions can you draw about your simulated data from this analysis?

An educational psychologist has developed a mediation technique to reduce anxiety. The psychologist selected a sample of high anxiety students that are asked to do the mediation at two therapy sessions a week apart. The participants' anxiety is measured the week before the first session and at each subsequent session. Below are the anxiety scores for the participants. What can the psychologist conclude with α= 0.05?

Make an interpretation based on the results.

At least one of the sessions differ on anxiety.None of the sessions differ on anxiety.    

e) Conduct Tukey's Post Hoc Test for the following comparisons:
2 vs. 3: difference =  ; significant:  ---Select--- Yes No
1 vs. 2: difference =  ; significant:  ---Select--- Yes No

f) Conduct Scheffe's Post Hoc Test for the following comparisons:
1 vs. 3: test statistic =  ; significant:  ---Select--- Yes No
2 vs. 3: test statistic =  ; significant:  ---Select--- Yes No

Module 3 Individual Problems

M3_IND1. A furniture cabinet maker produces two types of cabinets, Classic and Modern, that house and hide LCD TVs. The resource requirements and profit for the two types of cabinets are shown below.

800 12 350

The firm has a budget of $185,000 to spend on materials. The firm has 2,000 labor hours are available for use. What is the best combination of furniture cabinets to be made? Solve this two decision variable problem using the LP Graphing utility.

1. a) What is the profit (value of the objective function) for the optimal solution?

2. b) How many Classic models should be produced (based on the optimal solution)?

3. c) How many Modern models should be produced (based on the optimal solution)?

4. d) Is the production of 80 Classic models and 160 Modern models feasible (not asking if it is

   optimal). Does it fall in the feasible region?

5. e) Is the production of 160 Classic models and 100 Modern models feasible (not asking if it is

   optimal). Does it fall in the feasible region?

Resource Requirements and Profitability

Model

Classic Modern

Materials ($/unit) Labor (hrs./unit) Profit ($/unit)

600 4 250

1

M3_IND2. The fabrication department for an automobile component plant is scheduling its work for next month. The plant produces the following four components: A1, B2, C3, and D4. Each component must go through three departments during the fabrication process. After fabrication, each valve is inspected by a human being, who spends 15 minutes per valve. There are 500 inspection hours available for the month. The time required (in hours) for each department to work on each component is shown in the following table. Also shown are the minimum number of components that must be produced for the month and the unit profit for each component.

Formulate and solve the problem in Excel to determine the number of each product to manufacture that meets the requirements and maximizes profits.

1. a) What is the maximum profit based on your optimal solution (the value of the objective function)?

2. b) How many A1's should be manufactured based on your optimal solution?

3. c) How many B2's should be manufactured based on your optimal solution?

4. d) How many C3's should be manufactured based on your optimal solution?

5. e) How many D4's should be manufactured based on your optimal solution?

6. f) What is the total number of hours used in the drilling department based on your optimal

   solution?

7. g) What is the total number of hours used in the milling department based on your optimal

   solution?

8. h) What is the total number of hours used in the assembly department based on your optimal

   solution?

9. i) What is the total number of hours used in the inspection department based on your optimal

   solution?

2

M3_IND3. A snack company packages and sells three different canned party mixes that contain a total of 1 lb. of nuts. These three different products (Plain Nuts, Mixed Nuts, and Premium Mix) include a mix of four possible types of nuts (peanuts, cashews, almonds, and walnuts). The table below show the number of lbs. of each ingredient in each product type, the amount of ingredient available, and the revenue generated by selling each type of product. What should their production plan be to maximize their revenue? There is on additional piece of information that impacts their production plan and should be included in your formulation. Past demand indicates customers purchase at least three times as many cans of Plain Nuts as Mixed Nuts. Your formulation should include a constraint that states that the number of cans of Plain Nuts produced should be at least three times the number of cans of Mixed Nuts produced. Formulate and solve the problem in Excel to determine the number of each product to produce that meets the requirements and maximizes revenues. (Note: Consider this an average amount of cans produced – the number of cans does not need to be an integer).

0.8 0.25 500

0.2 0.25 0.2 300

0.25 0.4 120

0.25 0.4 100

$2.25 $5.65 $7.85

1. a) What is the maximum revenue based on your optimal solution (the value of the objective function)?

2. b) How many cans of Plain Nuts should be produced based on your optimal solution (enter two decimal places)?

3. c) How many cans of Mixed Nuts should be produced based on your optimal solution (enter two decimal places)??

4. d) How many cans of Premium Mix should be produced based on your optimal solution (enter two decimal places)??

5. e) After producing the number of cans of each product as suggested in your optimal solution, which of the ingredients has not been totally used by your production plan?

PRODUCT

INGREDIENTS

PEANUTS (lbs./can) CASHEWS (lbs./can)ALMONDS (lbs./can) WALNUTS (lbs./can) REVENUE ($/UNIT)

PLAIN MIXED PREMIUM INGREDIENT NUTS NUTS MIX AVAILABILITY (lbs.)

3

M3_IND4. A gear manufacturer is planning next week’s production for four types of gears. Becausethere are limited resources in the plant for production, the manufacturer can outsource the gears by purchasing these gears from a regional supplier. The regional supplier can supply a maximum of 400 units of each type of gear. The table below shows the exact demand for the gears, the revenue per unit, and the outsource cost per unit if the gears are purchased from the supplier. The manufacturer generates the same revenue per unit for the gears regardless of whether the gear is manufactured in their plant and then sold to their customers or outsourced from their supplier and then sold to their customers.

GEAR TYPE

Demand

RevenueOutsource Cost

GEAR A GEAR B GEAR C GEAR D

650

$13.75

$9.20

500

$12.50

$9.75

450

$16.90

$11.00

550

$18.50

$11.75

PRODUCT

When the gears are manufactured in the own plant, the gears must be processed through three different departments: forming, hardening, and deburring. The table below shows the processing time (in hours) for each type of gear in the departments as well as the capacity for each department and the cost per hour for processing the gears in those departments. The cost per hour for processing the gears is provided so that you can calculate the manufacturing cost.

0.30 0.25 0.31 0.40 400 $8.75

Formulate and solve this problem in Excel to determine the production and/or outsource plan which will meet the requirements and maximize the profit. (Hint: processing costs in the second table effect only the profit for the gears that are manufactured and not the gears that are outsourced)

1. a) How much profit does the company for all gears they make and buy in your solution (the value of the objective function)? (enter to the nearest integer)

2. b) If you could add one hour of capacity to any department to increase profit - adding one hour of capacity to which department would generate the biggest increase in profit: Forming, Hardening, or Deburring?

3. c) Which of the following constraints have slack? (Choose all constraints with slack): Forming, Hardening, or Deburring

4. d) In your solution, how many Gear C's should the company make?

5. e) If the cost per hour of the hardening process increases to $12/hr. - how many Gear D's should

   the company make with this new process cost?

PROCESS

GEAR A GEAR B (hrs./unit) (hrs./unit)

GEAR C GEAR D (hrs./unit) (hrs./unit)

DEPARTMENT CAPACITY (hours)

Forming Hardening Deburring

0.37 0.43

0.45 0.52 500

4

COST ($/hr.)

$9.50

0.40 0.37

0.42 0.32 400

$7.90

M3_IND5. An investor wishes to invest all of her $6.5 million in a diversified portfolio through a commercial lender. The types of investments, the expected annual interest rate for the investment, and the maximum allowed percentage of the total portfolio that the investment can represent are shown in the table below:

6.20% 25%

8.00% 25%

4.45% 30%

7.50% 15%

8.90% 10%

She wants at least 40% of her total investment in non-mortgage instruments. Furthermore, she wants no more than 35% of her total investment to be in high-yield and high-risk instruments (i.e. expected interest rate of investment is 8% or greater). Formulate and solve this problem in Excel to determine how her money should be diversified in a manner which will meet the requirements and maximize the amount of interest income. (Hint: Make sure that the LHS and RHS of constraints are the same units)

1. a) What is the expected total interest income generated from the investment strategy (the value of the objective function)?

2. b) Based on your solution, how much should be invested in government sponsored mortgage loans?

3. c) Based on your solution, how much should be invested in stock investments?

4. d) If you could increase the maximum allowed for the investments (in order to increase overall

   return) - which would you choose: conventional mortgage loans, bond investments, or

   governmental sponsored mortgage loans.

5. e) If the return on low-income mortgage loans was reduced to 4%, how much should be invested

   in these low-income mortgage loans based on your new solution?

INVESTMENT

EXPECTED INTEREST

MAXIMUM ALLOWED (% of total portfolio)

Low-income mortgage loans Conventional mortgage loans

7.40% 20%

Government sponsored

mortgage loans Bond investments Stock investments Futures trading

5

M3_IND6. A student project at WCU was initiated to try to determine the impact of implementation of new technologies. The students want to survey both distance and residential undergraduate students in the four different years at Western (first year, sophomore, junior, and senior). They have estimated that it will cost them $5.50 to survey first year and sophomore residential students and $8.00 to survey junior and senior residential students. The cost to interview distance students is slightly higher. It will cost $6.75 for first year and sophomores and $9.50 for junior and seniors. For statistical validity they want to interview at least 900 students. They feel that there are certain criteria that they must adhere to:

• At least 25% of first year students surveyed should be distance students

• At least 20% of sophomore students surveyed should be distance students

• At least 35% of junior students surveyed should be distance students

• At least 40% of senior students surveyed should be distance students

• No more than 35% of all the students surveyed should be first year students

• Juniors and seniors should be at least 45% of the students surveyed

• Each of the eight types of students must be represented in the survey by at least 10% of the

  total interviews

  Formulate and solve this problem in Excel to determine the number of each type of student that should be surveyed that meets the requirements and minimizes the cost to carry out the interviews.

1. a) What is the minimum cost in your optimal solution (the value of the objective function)?

2. b) If the cost of surveying first year and sophomore residential students increases from $5.50 to

   $7.00 – what is the new minimum cost in your optimal solution?

6

Binomial Distribution. Suppose that X has a binomial distribution with n = 50 and p = 0.6. Use Minitab to simulate 40 values of X.

MTB > random 40 c1;

SUBC > binomial 50 0.6.

Note: To find P(X < k) for any k > 0, use ‘cdf’ command; this works by typing:

MTB > cdf;

SUBC > binomial 50 0.6.

(a) What proportion of your values are less than 30? (b) What is the exact probability that X will be less than 30? (c) Find P(X < 28) and P(23 < X < 30) .

4. ( Just true or false for each, no need for explanation,thank you)

a.If factors being studied cannot be controlled, the data are said to be observational.True or False

b.After rejecting the null hypothesis of equal treatments, a researcher decided to compute a 95 percent confidence interval for the difference between the mean of treatment 1 and mean of treatment 2 based on Tukey's procedure. At α = .05, if the confidence interval includes the value of zero, then we can reject the hypothesis that the two population means are equal.True or False

c.The error sum of squares measures the between-treatment variability.True or False

d.In one-way ANOVA, a large value of F results when the within-treatment variability is large compared to the between-treatment variability.True or False

e.In one-way ANOVA, other factors being equal, the further apart the treatment means are from each other, the more likely we are to reject the null hypothesis associated with the ANOVA F test. True or False

1). Which predictor variables are statistically significant at the 10% significance level?

2). What is the slope and p-value of the bedrooms variable?

3). What percentage of the variability in price is explained by this model?