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Question & Answer:

  • a. Suppose that $68,000 is to be allocated for advertising, research, and investment in the ratio 8:6:3. How much money will be allocated for each
    b. Computer Warehouse sells batteries ($2) and small boxes of mechanical pencils ($6). In July, total sales were $1056. Customers bought 5 times as many batteries as boxes of mechanical pencils. How many of each did Computer Warehouse sell?
  • A quality characteristic of interest for a coffee-bag-filling process is the weight of the coffee in the individual bags. If the bags are under filled, two problems arise. First, customers may not be able to brew the coffee to be as strong as they wish. Second, the company may be in violation of the truth-in-labeling laws. In this example, the label weight on the package indicates that, on average, there are 5.45 grams of coffee in a bag. If the average amount of coffee in a bag exceeds the label weight, the company is giving away product. Getting an exact amount of coffee in a bag is problematic because of variation in the temperature and humidity inside the factory, differences in the density of the coffee, and the extremely fast filling operation of the machine (approximately 170 bags a minute). The following table provides the weight in grams of a sample of 50 bags produced in one hour by a single machine:



    a. Compute the arithmetic mean and median.
    b. Compute the first quartile and third quartile.
    c. Compute the range, interquartile range, variance, standard deviation, and coefficient of variation.
    d. Interpret the measures of central tendency within the context of this problem. Why should the company producing the coffee bags be concerned about the central tendency?
    e. Interpret the measures of variation within the context of this problem. Why should the company producing the coffee bags be concerned about variation?
  • An apple juice bottling company maintains records concerning the number of unacceptable bottles of juice obtained from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was nonconforming is 0.05 and the probability that a bottle came from machine II and was nonconforming is 0.075. These probabilities represent the probability of one bottle out of the total sample having the specified characteristics. Half the bottles are filled on machine I and the other half are filled on machine II.

    a. If a filled bottle of juice is selected at random, what is the probability that it is a nonconforming bottle?
    b. If a filled bottle of juice is selected at random, what is the probability that it was filled on machine II?
    c. If a filled bottle of juice is selected at random, what is the probability that it was filled on machine I and is a conforming bottle?
  • According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 24.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 6.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.

    a. Find the probability that the return for common stocks will be greater than 0%.
    b. Find the probability that the return for common stocks will be less than 20%.
  • Compute a 95% confidence interval for the population mean, based on the sample 10, 12, 13, 14, 15, 16, and 49. Change the number from 49 to 16 and recalculate the confidence interval. Using the results, describe the effect of an outlier or extreme value on the confidence interval
  • The director of admissions at the University of Maryland, University College is concerned about the high cost of textbooks for the students each semester. A sample of 25 students enrolled in the university indicates that X (bar) = $315.4 and s = $43.20.

    a. Using the 0.10 level of significance, is there evidence that the population mean is above $300?
    b. What is your answer in (a) if s = $75 and the 0.05 level of significance is used?
    c. What is your answer in (a) if X (bar) = $305.11 and s = $43.20?
    d. Based on the information in part (a), what decision should the director make about the books used for the courses if the goal is to keep the cost below $300?
  • A large candy manufacturer is concerned that the mean weight of their bag of Gooey Sour Worms is not greater than 7.3 ounces. It can be assumed that the population standard deviation is .5 ounces based on past experience. A sample of 169 gummy worms is selected and the sample mean is 7.35 ounces. Using a level of significance of .10, is there evidence that the population mean weight of the candy bars is greater than 7.3? Fully explain your answer.
  • Each of the possible five outcomes of a random experiment is equally likely. The sample space is {a, b, c, d, e}.
    Let A denote the event {a, b}, and let B denote the event {c, d, e}. Determine the following:
    (a) P (A) (b) P (B)
    (c) P (A’) (d) P (A U B)
    (e) P (A ∩ B)
  • The sample space of a random experiment is {a, b, c, d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. Let A denote the event {a, b, c}, and let B denote the event
    {c, d, e}. Determine the following:
    (a) P (A) (b) P (B)
    (c) P (A’) (d) P (A U B)
    (e) P (A ∩ B)
  • A part selected for testing is equally likely to have been produced on any one of six cutting tools.
    (a) What is the sample space?
    (b) What is the probability that the part is from tool 1?
    (c) What is the probability that the part is from tool 3 or tool 5?
    (d) What is the probability that the part is not from tool 4?
  • An injection-molded part is equally likely to be obtained from any one of the eight cavities on a mold.
    (a) What is the sample space?
    (b) What is the probability a part is from cavity 1 or 2?
    (c) What is the probability that a part is neither from cavity 3 nor 4?
  • A sample space contains 20 equally likely outcomes.
    If the probability of event A is 0.3, how many outcomes are in event A?
  • Orders for a computer are summarized by the optional features that are requested as follows:
    Proportion of orders
    No optional features 0.3
    One optional feature 0.5
    More than one optional feature 0.2
    (a) What is the probability that an order requests at least one optional feature?
    (b) What is the probability that an order does not request more than one optional feature?
  • If the last digit of a weight measurement is equally likely to be any of the digits 0 through 9,
    (a) What is the probability that the last digit is 0?
    (b) What is the probability that the last digit is greater than or equal to 5?
  • A sample preparation for a chemical measurement is completed correctly by 25% of the lab technicians, completed with a minor error by 70%, and completed with a major error by 5%.
    (a) If a technician is selected randomly to complete the preparation, what is the probability it is completed without error?
    (b) What is the probability that it is completed with either a minor or a major error
  • A credit card contains 16 digits between 0 and 9. However, only 100 million numbers are valid. If a number is entered randomly, what is the probability that it is a valid number?
  • Suppose your vehicle is licensed in a state that issues license plates that consist of three digits (between 0 and 9) followed by three letters (between A and Z). If a license number is selected randomly, what is the probability that yours is the one selected
  • A message can follow different paths through servers on a network. The senders message can go to one of five servers for the first step, each of them can send to five servers at the second step, each of which can send to four servers at the third step, and then the message goes to the recipients server.
    (a) How many paths are possible?
    (b) If all paths are equally likely, what is the probability that a message passes through the first of four servers at the third step?
  • Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:
    Shock resistance
    High Low
    Scratch high 70 9
    Resistance low 16 5
    Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance.
    If a disk is selected at random, determine the following probabilities:
    (a) P (A) (b) P (A)
    (c) P (A`) (d) P (A ∩ B)
    (e) (A U B) (f) (A` U B)
  • Samples of a Fast aluminum part are classified on the basis of surface finish (in micro inches) and edge finish. The results of 100 parts are summarized as follows:
    Edge finish
    Excellent Good
    Surface excellent 80 2
    Finish good 10 8
    Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent length. If a part is selected at random,
    determine the following probabilities:
    (a) P (A) (b) P (A)
    (c) P (A`) (d) P (A ∩ B)
    (e) (A U B) (f) (A` U B)