# initial screening | Mathematics homework help

**Question Detail:**

1. SeaFair Fashions relies on its sales force of 220 to do an initial screening of all new fashions. The company is presently bringing out a new line of swimwear and has invited

40 salespeople to its Orlando home office. An issue of constant concern to the SeaFair sales office is the volume of orders generated by each salesperson. Last year the overall company average was $417,330 with a standard deviation of $45,285.

- What shape do you think the distribution of all possible sample means of 40 will have? Discuss.
- Determine the value of the standard deviation of the distribution of the sample mean of all possible samples of size 40.

c. Determine the probability that the sample of 40 will have a sales average less than

$400,000.

- How would the answer to part a, b, and c change if the home office brought 60 salespeople to Orlando? Provide the respective answers for this sample size.

e. Each year SeaFair invites the sales personnel with sales above the 85th percentile to enjoy a complimentary vacation in Hawaii. Determine the smallest sales level for the salespersonnel who were awarded a trip to Hawaii last year. (Assume the distribution of sales was normally distributed last year).

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2. Tom Marley and Jennifer Griggs have recently started a marketing research firm in Jacksonville, Florida. They have contacted the Florida Democratic Party with a proposal to do allpolitical polling for the party. Since they have just started their company, the state party chairman is reluctant to sign a contract without some test of their accuracy. He has asked them todo a trial poll in a central Florida county known to have 60% registered Democratic Party voters. The poll itself had many questions. However, for the test of accuracy, only the proportionof registered Democrats was considered. Tom and Jennifer report back that from a random sample of 760 respondents, 395 were registered Democrats.

**a.**Determine the probability that such a random sample would result in 395 or fewer

Democrats in the sample.

**b.**Based on your calculations in part a, would you recommend that the Florida Democratic Party (or anyone else for that matter) contract with the Marley/Griggs marketing research firm?Explain your answer.

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3. Micron Electronics makes personnel computers that are then sold directly over the phone and over the Internet. One of the most critical factors in the success of PC makers is how fastthey can turn their inventory of parts. Faster inventory turns means lower average inventory cost. Recently at a meeting, the VP of manufacturing said that there is no reason to continueoffering hard disk drives that have less than a 2.0 GB storage capacity since only 10% of Micron customers ask for the smaller hard disks. After much discussion and debate about theaccuracy of the VP’s figure, a sample of 100 orders from the past week’s sales was taken. This sample revealed 14 requests for drives with less than 2.0 GB capacity.

**a.**Determine the probability of finding 14 or more requests for hard disk drives with low storage capacity if the VP’s assertion is correct. Do you believe that the proportion ofcustomers requesting hard drives with low storage capacity is smaller than 0.10? Explain.

**b.**Suppose a second sample of 100 customers was selected. This sample again yielded 14 requests for less than a 2 GB drive. Combining this sample information with that in part a,what conclusion would you now reach regarding the VP’s 10% claim? Base your answer on probability.

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4. Agri-Beef, Inc. is a large Midwestern farming operation. The company has been a leader in employing statistical techniques in its business. Recently, John Goldberg, operationsmanager, requested that a random sample of cattle be selected and that these cattle be fed a special diet. The cattle were weighed before the start of the new feeding program and at theend of the feeding program. John wished to estimate the average daily weight gain for cattle on the new feed program. Two hundred cattle were tested,

with the following sample results: X = 1.2 pounds gain per day and S= 0.50 pounds gain per day.

a. Obtain a 95% confidence interval estimate for the true average daily weight gain.

b. Provide a 90% confidence interval estimate for the true average daily weight gain.

c. Discuss the difference between the two estimates found in parts a and b, and indicate the advantages and disadvantages of each.

d. John is considering adopting this new diet. However, the weight gain comes at a price. To feed 200 cows for one month, the diet would cost approximately $1,000 more than theircurrent feed program. If the price of beef on the hoof has been close to $0.20 a pound, would such a program be cost effective for Agri-Beef? Support your answer with calculations and statistical reasoning.

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- A major American pharmaceutical company has randomly sampled 14 customers who have had prescriptions for one of their new pain killing drugs for 2 months. There is concern thatthe drug may elevate the user’s heart rate. Each of the customers in the sample had his or her heart rate measured after using the drug for 1 week. All people in the sample had heart ratesof 55 prior to taking the drug. The following data were recorded for the 14 customers:

50 70 60 70 90 72 50

80 85 55 66 70 80 40

a. Suppose that you have just started working in the marketing department of the pharmaceutical company. You were given the following instructions: “Based on these sample dataconstruct a 90% confidence interval estimate for the true mean heart rate for the company’s drug customers. Interpret the estimate.” (Assume that heart rates are normally distributed.)

b. Referring to your answer in part a, could the estimate be applied to all potential drug customers? Explain why or why not.

**c.**Refer to your calculations in part a. Was the concern expressed justified? Justify your answer. If you conclude the average heart rate did not increase, determine the probability that asample mean at least as large as the one obtained in your sample could actually have been obtained.

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6. Arco Manufacturing makes electronic pagers. As part of the company’s quality efforts, the company wishes to estimate the mean number of days the pager is used before repair isneeded. A pilot sample of 40 pagers indicates a sample standard deviation of

200 days. The company wishes its estimate to have a margin of error of no more than 50 days and the confidence level must be 95%.

**a.**Given this information, how many additional pagers should be sampled?

**b.**The pilot study was initiated because of the costs involved in sampling. Each

sampled observation costs approximately $10 to obtain. Originally, it was thought that the population’s standard deviation may be as large as 300. Determine the amount of money saved by obtaining the pilot sample.

7. A local radio station is interested in estimating the percentage of a target market that has a favorable impression of its morning show. The marketing department wishes to estimate thisproportion within ±0.03 of the true population value and have a confidence level of 98%. A pilot sample of 40 people was selected, and the proportion in this sample with a favorable impression was 0.45.

a. Based on this information, how many more people must be surveyed?

**b.**Determine the margin of error in the pilot sample. Assume a confidence level of

98%.

8. Refer to problem 7. Suppose the station managers can’t afford to survey as many people as required and have indicated that the maximum sample size can be 700 people. Assuming that the sample size is cut to 700:

**a.**Without changing the confidence level, indicate specifically what must be changed and by how much.

**b.**Without changing the margin of error, indicate specifically what must be changed and by how much.

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9. Most major airlines allow passengers to carry two pieces of luggage (of a certain maximum size) onto the plane. However, their studies show that the more carry-on baggagepassengers have, the longer it takes the plane to unload and load passengers. One regional airline is considering changing its policy to allow only one carry-on per passenger. Before doingso, the airline decided to collect some data. Specifically, a random sample of 1,000 passengers was selected. Researchers observed the passengers and noted the number of bags eachperson carried on the plane. Out of the 1,000 passengers, 345 had more than one bag.

a. Based on this sample, develop and interpret a 95% confidence interval estimate for the proportion of the traveling population that would have been impacted had the “one-bag” limit been in effect. Discuss your result.

b. The domestic version of Boeing’s 747 has a capacity for 568 passengers.

Determine an interval estimate of the number of passengers you would expect to board the plane with more than one carry-on. Assume the plane is at its passenger capacity.

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10. A telemarketing company located in Los Angeles has established a guideline stating that the average time for each completed call should be no more than 4 minutes. Recently theoperations manager was concerned that calls were taking too long. The operations manager did not wish to assert that the calls were taking too long if the sample data did not stronglyindicate this. A sample of 12 calls was selected and the following times (in seconds) were recorded.

194 278 302 140 245 234 268 208 302 190 320 255

**a.**Construct the appropriate null and alternative hypotheses.

**b.**Based on the sample data, what should the operations manager conclude? Test at the 0.10 significance level using the p-value approach. (Assume that call times are normally distributed.)

**c.**Suppose you wished to conduct the test in part b using X as the test statistic.

Calculate the critical value of X.

**d.**Consider your answer to part c. What values of the population mean could have been specified in the null hypothesis so that you would not reject the null hypothesis?

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- A mail-order business prides itself in its ability to fill customers’ orders in 6 calendar days or less on the average. Periodically, the operations manager selects a random sample of customer orders and determines the number of days required to fill their orders. Based upon this sample information, he decides whether the desired standard is being met. He will assumethat the average number of days to fill customer orders is 6 or less unless the data strongly suggest otherwise.

a. Establish the appropriate null and alternative hypotheses.

b. On one occasion where a sample of 40 customers was selected, the average number of days was 6.65, with a sample standard deviation of 1.5 days. Can the operations managerconclude that this mail-order business is achieving its goals? **Explain**. Use a significance level of 0.025 to answer this question.