| № |
Condition |
free/or 0.5$ |
| m845 | Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows:
Conforms
Yes No
1 22 8
Supplier 2 25 5
3 30 10
Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. If a sample 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) |
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| m846 | 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) |
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| m847 | Samples of laboratory glass are in small, light packaging or heavy, large packaging. Suppose that 2 and 1% of the sample shipped in small and large packages, respectively, break during transit. If 60% of the samples are shipped in large packages and 40% are shipped in small packages, what proportion of samples break during shipment? |
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| m848 | Samples of size n = 4 are selected from a production line.
a. What is the value of the A2 factor used to determine the upper and lower control limits for the mean?
b. What are the values of the D3 and D4 factors used to determine the lower and upper control limits for the range? |
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| m849 | Sam’s Supermarkets tests its checkout clerks by randomly examining the printout receipts for scanning errors. The following numbers are the number of errors on each receipt for October 27: 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0. Construct a control chart for this process and comment on whether the process is “in control.” |
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| m850 | SCCoast, an Internet provider in the Southeast, developed the following frequency distribution on the age of Internet users. Estimate the mean and the standard deviation.
Age (years) Frequency
10 up to 20......... 3
20 up to 30......... 7
30 up to 40......... 18
40 up to 50 ......... 20
50 up to 60......... 12 |
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| m851 | Scrapper Elevator Company has 20 sales representatives who sell its product throughout the United States and Canada. The number of units sold last month by each representative is listed below. Assume these sales figures to be the population values.
a. Draw a graph showing the population distribution.
b. Compute the mean of the population.
c. Select five random samples of 5 each. Compute the mean of each sample. Use the methods described in this chapter and Appendix B.4 to determine the items to be included in the sample.
d. Compare the mean of the sampling distribution of the sample mean to the population mean. Would you expect the two values to be about the same?
e. Draw a histogram of the sample means. Do you notice a difference in the shape of the distribution of sample means compared to the shape of the populationdistribution? |
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| m852 | Semiconductor lasers used in optical storage products require higher power levels for write operations than for read operations. High-power-level operations lower the useful life of the laser. Lasers in products used for backup of higher speed magnetic disks primarily write, and the probability that the useful life exceeds five years is 0.95. Lasers that are in products that are used for main storage spend approximately an equal amount of time reading and writing, and the probability that the useful life exceeds five years is 0.995. Now, 25% of the products from a manufacturer are used for backup and 75% of the products are used for main storage. Let A denote the event that a laser’s useful life exceeds five years, and let B denote the event that a laser is in a product that is used for backup. Use a tree diagram to determine the following:
(a) P (B)
(b) P (A|B)
(c) P (A|B’)
(d) P (A ∩ B)
(e) P (A ∩ B’)
(f) P (A)
(g) What is the probability that the useful life of a laser exceeds five years?
(h) What is the probability that a laser that failed before five years came from a product used for backup? |
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| m853 | Shank’s Inc., a nationwide advertising firm, wants to know whether the size of an advertisement and the color of the advertisement make a difference in the response of magazine readers. A random sample of readers is shown ads of four different colors and three different sizes. Each reader is asked to give the particular combination of size and color a rating between 1 and 10. Assume that the ratings follow the normal distribution. The rating for each combination is shown in the following table (for example, the rating for a small red ad is 2).
Is there a difference in the effectiveness of an advertisement by color and by size? Use the .05 level ofsignificance. |
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| m854 | Shaver Manufacturing Inc. offers dental insurance to its employees. A recent study by the human resource director shows the annual cost per employee per year followed the normal probability distribution, with a mean of $1,280 and a standard deviation of $420 per year.
a. What fraction of the employees costs more than $1,500 per year for dental expenses?
b. What fraction of the employees costs between $1,500 and $2,000 per year?
c. Estimate the percent that did not have any dental expense.
d. What was the cost for the 10% of employees who incurred the highest dental expense? |
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| m855 | Shaver Manufacturing Inc. offers dental insurance to its employees. A recent study by the human resource director shows the annual cost per employee per year followed the normal probability distribution, with a mean of $1,280 and a standard deviation of $420 per year.
a. What fraction of the employees costs more than $1,500 per year for dental expenses?
b. What fraction of the employees costs between $1,500 and $2,000 per year?
c. Estimate the percent that did not have any dental expense.
d. What was the cost for the 10% of employees who incurred the highest dental expense? |
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| m856 | A shipment of chemicals arrives in 15 totes. Three of the totes are selected at random, without replacement, for an inspection of purity. If two of the totes do not conform to purity requirements, what is the probability that at least one of the nonconforming totes is selected in the sample? |
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| m857 | Shorty’s Muffler advertises it can install a new muffler in 30 minutes or less. However, the work standards department at corporate headquarters recently conducted a study and found that 20% of the mufflers were not installed in 30 minutes or less. The Maumee branch installed 50 mufflers last month. If the corporate report is correct:
a. How many of the installations at the Maumee branch would you expect to take more than 30 minutes?
b. What is the likelihood that fewer than eight installations took more than 30 minutes?
c. What is the likelihood that eight or fewer installations took more than 30 minutes?
d. What is the likelihood that exactly 8 of the 50 installations took more than 30 minutes? |
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| m858 | Show by induction on n that R = [a1, b1] x..x [an, bn] is not a set of measure 0 (or content 0) if ai < bi for each i. |
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| m859 | Show that for a discrete uniform random variable X, if each of the values in the range of X is multiplied by the constant c, the effect is to multiply the mean of X by c and the variance of X by c2. That is, show that (cX) = cE (X) and V(cX) = c2V(X). |
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| m860 | Show that if C has content 0, then C C A for some closed rectangle A and C is Jordan-measurable and ∫ AXC = 0. |
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| m861 | Show that if f, g: A -> R are integrable, so is f ∙ g. |
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| m862 | Show that Mx consists of the tangent vectors at t of curves in M with c (t) = x. |
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| m863 | Show that the continuity of D1 f j at a may be eliminated from the hypothesis of Theorem 2-8. |
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| m864 | Show that the function f (x) in Example 3-5 satisfies the properties of a probability mass function by summing the infinite series. |
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