# Quantitative Methods

**MOD 4**

**Start by reading and following these instructions:**

1. Quickly skim the questions or assignment below and the assignment rubric to help you focus.

2. Read the required chapter(s) of the textbook and any additional recommended resources. Some answers may require you to do additional research on the Internet or in other reference sources. Choose your sources carefully.

3. Consider the discussion and the any insights you gained from it.

4. Create your Assignment submission and be sure to cite your sources, use APA style as required, check your spelling.

**Assignment:**

*Any documents needed to complete the assignment questions are attached below* *.*

1. To satisfy concerns of potential customers, the management of OurCampus! has undertaken a research project to learn the amount of time it takes users to load a complex video features page. The research team has collected data and has made some claims based on the assertion that the data follow a normal distribution. Open, which documents the work of a quality response team at OurCampus! Read the internal report that documents the work of the team and their conclusions. Then answer the following:

a. Can the collected data be approximated by the normal distribution?

b. Review and evaluate the conclusions made by the OurCampus! research team. Which conclusions are correct? Which ones are incorrect?

c. If OurCampus! could improve the mean time by five seconds, how would the probabilities change?

2. Toss a coin 10 times and record the number of heads. If each student performs this experiment five times, a frequency distribution of the number of heads can be developed from the results of the entire class. Does this distribution seem to approximate the normal distribution?

3. The advocacy group Consumers Concerned About Cereal Cheaters (CCACC) suspects that cereal companies, including Oxford Cereals, are cheating consumers by packaging cereals at less than labeled weights. Recently, the group investigated the package weights of two popular Oxford brand cereals. Open CCACC.pdf to examine the group’s claims and supporting data, and then answer the following questions:

a. Are the data collection procedures that the CCACC uses to form its conclusions flawed? What procedures could the group follow to make its analysis more rigorous?

b. Assume that the two samples of five cereal boxes (one sample for each of two cereal varieties) listed on the CCACC website were collected randomly by organization members. For each sample, do the following:

i. Calculate the sample mean.

ii. Assume that the standard deviation of the process is 15 grams and the population mean is 368 grams. Calculate the percentage of all samples for each process that have a sample mean less than the value you calculated in (i).

iii. Again, assuming that the standard deviation is 15 grams, calculate the percentage of individual boxes of cereal that have a weight less than the value you calculated in (i).

c. What, if any, conclusions can you form by using your calculations about the filling processes for the two different cereals?

d. A representative from Oxford Cereals has asked that the CCACC take down its page discussing shortages in Oxford Cereals boxes. Is that request reasonable? Why or why not?

e. Can the techniques discussed in this chapter be used to prove cheating in the manner alleged by the CCACC? Why or why not?

4. Using Random Number Table E.1 from Page 544 of the textbook, simulate the selection of different-colored balls from a bowl, as follows:

a. Start in the row corresponding to the day of the month in which you were born.

b. Select one-digit numbers.

c. If a random digit between 0 and 6 is selected, consider the ball white; if a random digit is a 7, 8, or 9, consider the ball red.

d. Select samples of digits. In each sample, count the number of white balls and compute the proportion of white balls in the sample. If each student in the class selects five different samples for each sample size, a frequency distribution of the proportion of white balls (for each sample size) can be developed from the results of the entire class. What conclusions can you reach about the sampling distribution of the proportion as the sample size is increased?

e. Suppose that step c of this problem uses the following rule: “If a random digit between 0 and 8 is selected, consider the ball to be white; if a random digit of 9 is selected, consider the ball to be red.” Compare and contrast the results in this problem and those in step c.

5. The fill amount of bottles of a soft drink is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liter. If you select a random sample of 25 bottles, what is the probability that the sample mean will be

a. between 1.99 and 2.0 liters?

b. below 1.98 liters?

c. greater than 2.01 liters?

d. The probability is 99% that the sample mean amount of soft drink will be at least how much?

e. The probability is 99% that the sample mean amount of soft drink will be between which two values (symmetrically distributed around the mean)?

**QUAN METHODS**

**Start by reading and following these instructions:**

1. Quickly skim the questions or assignment below and the assignment rubric to help you focus.

2. Read the required chapter(s) of the textbook and any additional recommended resources. Some answers may require you to do additional research on the Internet or in other reference sources. Choose your sources carefully.

3. Consider the discussion and the any insights you gained from it.

4. Create your Assignment submission and be sure to cite your sources, use APA style as required, check your spelling.

**Assignment:**

*Any documents needed to complete the assignment questions are attached below* *.*

1. Among its other features, the OurCampus! website allows customers to purchase OurCampus! LifeStyles merchandise on-line. To handle payment processing, the management of OurCampus! has contracted with the following firms:

a. PayAFriend (PAF) This is an online payment system with which customers and businesses such as OurCampus! register in order to exchange payments in a secure and convenient manner, without the need for a credit card.

b. Continental Banking Company (Conbanco) This processing services provider allows OurCampus! customers to pay for merchandise using nationally recognized credit cards issued by a financial institution.

To reduce costs, management is considering eliminating one of these two payment systems. However, Lorraine Hildick of the sales department suspects that customers use the two forms of payment in unequal numbers and that customers display different buying behaviors when using the two forms of payment. Therefore, she would like to first determine the following:

a. The proportion of customers using PAF and the proportion of customers using a credit card to pay for their purchases.

b. The mean purchase amount when using PAF and the mean purchase amount when using a credit card.

Assist Ms. Hildick by preparing an appropriate analysis. Download the Payments Sample PDF provided in Module 5, read Ms. Hildick’s comments, and use her random sample of 50 transactions as the basis for your analysis. Summarize your findings to determine whether Ms. Hildick’s conjectures about OurCampus! customer purchasing behaviors are correct. If you want the sampling error to be no more than $3 when estimating the mean purchase amount, is Ms. Hildick’s sample large enough to perform a valid analysis?

2. In response to the negative statements made by the Concerned Consumers About Cereal Cheaters (CCACC) in the module 5 Digital Case, Oxford Cereals recently conducted an experiment concerning cereal packaging. The company claims that the results of the experiment refute the CCACC allegations that Oxford Cereals has been cheating consumers by packaging cereals at less than labeled weights. Download the Oxford Current News PDF provided in Module 5 that provides a portfolio of current news releases from Oxford Cereals. Review the relevant press releases and supporting documents.

Then answer the following questions:

a. Are the results of the experiment valid? Why or why not? If you were conducting the experiment, is there anything you would change?

b. Do the results support the claim that Oxford Cereals is not cheating its customers?

c. Is the claim of the Oxford Cereals CEO that many cereal boxes contain more than 368 grams surprising? Is it true?

d. Could there ever be a circumstance in which the results of the Oxford Cereals experiment and the CCACC’s results are both correct? Explain

3. A manufacturing company produces electrical insulators. If the insulators break when in use, a short circuit is likely to occur. To test the strength of the insulators, destructive testing is carried out to determine how much force is required to break the insulators. Force is measured by observing the number of pounds of force applied to the insulator before it breaks.

Use the data stored in Force.xls (from 30 insulators) and subject it to this testing:

a. At the 0.05 level of significance, is there evidence that the population mean force required to break the insulator is greater than 1,500 pounds?

b. What assumption about the population distribution is needed in order to conduct the t test in (a)?

c. Construct a histogram, boxplot, or normal probability plot to evaluate the assumption made in (b).

d. Do you think that the assumption needed in order to con-duct the t test in (a) is valid? Explain.

4. The manufacturer of Boston and Vermont asphalt shingles provides its customers with a 20-year warranty on most of its products. To determine whether a shingle will last through the warranty period, accelerated-life testing is conducted at the manufacturing plant. Accelerated-life testing exposes the shingle to the stresses it would be subject to in a lifetime of normal use in a laboratory setting via an experiment that takes only a few minutes to conduct. In this test, a shingle is repeatedly scraped with a brush for a short period of time, and the shingle granules removed by the brushing are weighed (in grams). Shingles that experience low amounts of granule loss are expected to last longer in normal use than shingles that experience high amounts of granule loss. The file Granule contains a sample of 170 measurements made on the company’s Boston shingles and 140 measurements made on Vermont shingles.

a. For the Boston shingles, is there evidence that the population mean granule loss is different from 0.50 grams?

b. Interpret the meaning of the p-value in (a).

c. For the Vermont shingles, is there evidence that the population mean granule loss is different from 0.50 grams?

d. Interpret the meaning of the p-value in (c).

e. In (a) through (d), do you have to worry about the normality assumption? Explain.

5. An important quality characteristic used by the manufacturer of Boston and Vermont asphalt shingles is the amount of moisture the shingles contain when they are packaged. Customers may feel that they have purchased a product lacking in quality if they find moisture and wet shingles inside the packaging. In some cases, excessive moisture can cause the granules attached to the shingle for texture and coloring purposes to fall off the shingle, resulting in appearance problems. To monitor the amount of moisture present, the company conducts moisture tests. A shingle is weighed and then dried. The shingle is then reweighed, and, based on the amount of moisture taken out of the product, the pounds of moisture per 100 square feet are calculated. The company would like to show that the mean moisture content is less than 0.35 pound per 100 square feet. The file Moisture includes 36 measurements (in pounds per 100 square feet) for Boston shingles and 31 for Vermont shingles.

a. For the Boston shingles, is there evidence at the 0.05 level of significance that the population mean moisture content is less than 0.35 pound per 100 square feet?

b. Interpret the meaning of the p-value in (a).

c. For the Vermont shingles, is there evidence at the 0.05 level of significance that the population mean moisture content is less than 0.35 pound per 100 square feet?

d. Interpret the meaning of the p-value in (c).

e. What assumption about the population distribution is needed in order to conduct the t tests in (a) and (c)?

f. Construct histograms, boxplots, or normal probability plots to evaluate the assumption made in (a) and (c).

g. Do you think that the assumption needed in order to conduct the t tests in (a) and (c) is valid? Explain.

**MOD 6**

**Start by reading and following these instructions:**

1. Quickly skim the questions or assignment below and the assignment rubric to help you focus.

2. Read the required chapter(s) of the textbook and any additional recommended resources. Some answers may require you to do additional research on the Internet or in other reference sources. Choose your sources carefully.

3. Consider the discussion and the any insights you gained from it.

4. Create your Assignment submission and be sure to cite your sources, use APA style as required, check your spelling.

**Assignment:**

*Any documents needed to complete the assignment questions are attached below* *.*

1. **Part 1**

Even after the recent public experiment about cereal box weights, Consumers Concerned About Cereal Cheaters (CCACC) remains convinced that Oxford Cereals has misled the public. The group has created and circulated MoreCheating.pdf, a document in which it claims that cereal boxes produced at Plant Number 2 in Springville weigh less than the claimed mean of 368 grams.

Review this document and then answer the following questions:

a. Do the CCACC’s results prove that there is a statistically significant difference in the mean weights of cereal boxes produced at Plant Numbers 1 and 2?

b. Perform the appropriate analysis to test the CCACC’s hypothesis. What conclusions can you reach based on the data?

**Part 2**

After reviewing the CCACC’s latest document, Oxford Cereals has released SecondAnalysis.pdf, a press kit that Oxford Cereals has assembled to refute the claim that it is guilty of using selective data.

Review the Oxford Cereals press kit and then answer the following questions:

c. Does Oxford Cereals have a legitimate argument? Why or why not?

d. Assuming that the samples Oxford Cereals has posted were randomly selected, perform the appropriate analysis to resolve the ongoing weight dispute.

e. What conclusions can you reach from your results? If you were called as an expert witness, would you support the claims of the CCACC or the claims of Oxford Cereals? Explain.

2. The data in represent the 3-year annualized return, 5-year annualized return, 10-year annualized return, and expense ratio (in %) for the 10 mutual funds rated best by the U. S. News & World Report for intermediate municipal bond, short- term bond, and intermediate-term bond categories. (Data extracted from K. Shinkle, “The Best Funds for the Long Term, U. S. News & World Report, Summer 2010, pp. 52– 56.) Analyze the data and determine whether any differences exist between intermediate municipal bond, short-term bond, and intermediate- term bond mutual funds. (Use the 0.05 level of significance.)

3. An article (A. Jennings, “What’s Good for a Business Can Be Hard on Friends,” The New York Times, August 4, 2007, pp. C1– C2) reported that according to a poll, the mean number of cell phone calls per month was 290 for 18-24-year-olds and 194 for 45-54-year-olds, whereas the mean number of text messages per month was 290 for 18-24-year-olds and 57 for 45-54-year-olds. Suppose that the poll was based on a sample of 100 18-24-year-olds and 100 45-54-year-olds and that the standard deviation of the number of cell phone calls per month was 100 for 18-24-year-olds and 90 for 45-54-year-olds, whereas the standard deviation of the number of text messages per month was 90 for 18-24-year-olds and 77 for 45-54-year-olds. Assume a level of significance of 0.05.

a. Is there evidence of a difference in the variances of the number of cell phone calls per month for 18- 24-year-olds and for 45-54-year-olds?

b. Is there evidence of a difference in the mean number of cell phone calls per month for 18-24-year-olds and for 45-54-year-olds?

c. Construct and interpret a 95% confidence interval estimate for the difference in the mean number of cell phone calls per month for 18-24-year-olds and 45-54-year-olds.

d. Is there evidence of a difference in the variances of the number of text messages per month for 18-24-year-olds and 45-54-year-olds?

e. Is there evidence of a difference in the mean number of text messages per month for 18-24-year-olds and 45-54-year-olds?

f. Construct and interpret a 95% confidence interval estimate for the difference in the mean number of text messages per month for 18-24-year-olds and 45-54-year-olds.

g. Based on the results of (a) through (f), what conclusions can you make concerning cell phone and text message usage between 18-24-year-olds and 45-54-year-olds?

4. As T. C. Resort Properties seeks to improve its customer service, the company faces new competition from SunLow Resorts. SunLow has recently opened resort hotels on the islands where T. C. Resort Properties has its five hotels. SunLow is currently advertising that a random survey of 300 customers revealed that about 60% of the customers preferred its “Concierge Class” travel reward program over the T. C. Resorts “TCRewards Plus” program. Open and review ConciergeClass.pdf, an electronic brochure that describes the Concierge Class program and compares it to the T. C. Resorts program.

Then answer the following questions:

a. Are the claims made by SunLow valid?

b. What analyses of the survey data would lead to a more favorable impression about T. C. Resort Properties?

c. Perform one of the analyses identified in your answer to step 2.

d. Review the data about the T. C. Resorts properties customers presented in this chapter. Are there any other questions that you might include in a future survey of travel reward programs? Explain.

5. Phase 1

Reviewing the results of its research, the marketing department team concluded that a segment of Ashland households might be interested in a discounted trial subscription to the AMS 3-For-All cable/phone/Internet service. The team decided to test various discounts before determining the type of discount to offer during the trial period. It decided to conduct an experiment using three types of discounts plus a plan that offered no discount during the trial period:

a. No discount for the 3-For-All cable/ phone/ Internet service. Subscribers would pay $24.99 per week for the 3-For-All cable/ phone/ Internet service during the 90-day trial period.

b. Moderate discount for the 3-For-All cable/phone/Internet service. Subscribers would pay $19.99 per week for the 3-For-All cable/phone/Internet service during the 90-day trial period.

c. Substantial discount for the 3-For-All cable/phone/Internet service. Subscribers would pay $14.99 per week for the 3-For-All cable/phone/Internet service during the 90-day trial period.

d. Discount restaurant card. Subscribers would be given a Gold card providing a discount of 15% at selected restaurants in Ashland during the trial period. Each participant in the experiment was randomly assigned to a discount plan. A random sample of 100 subscribers to each plan during the trial period was tracked to determine how many would continue to subscribe to the 3-For-All service after the trial period. Table in Figure 10 summarizes the results.

TABLE in Figure 10 Number of Subscribers Who Continue Subscriptions After Trial Period with Four Discount Plans

Figure 10

Analyze the results of the experiment. Write a report to the team that includes your recommendation for which discount plan to use. Be prepared to discuss the limitations and assumptions of the experiment.

Phase 2

The marketing department team discussed the results of the survey presented in Module 6, in the Discussion Question. The team realized that the evaluation of individual questions was providing only limited information. In order to further understand the market for the 3-For-All cable/ phone/ Inter-net service, the data were organized in the following contingency tables:

Figure 11

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Figure 12

Figure 13

Analyze the results of the contingency tables. Write a report for the marketing department team and discuss the marketing implications of the results for Ashland MultiComm Services.

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