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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

Following Exercises require the use of counting techniques. 1. A committee of three people is to be selected at random from a county council consisting of five Democrats and four Republicans. What is the expected number of Democrats on the committee? Republicans? 2. Four balls are selected at
1. What is the expected number of red balls when three balls are selected at random, with replacement, from an urn containing four red balls and three green balls? Without replacement? 2. Cards What is the expected number of hearts when two cards are selected at random, with replacement, from a
1. The promoter of a football game is concerned that it will rain. They have the option of spending $8000 on insurance that will pay $40,000 if it rains. They estimate that the revenue from the game will be $60,000 if it does not rain and $25,000 if it does rain. What must the chance of rain be if
1. An assembly line produces 12 bicycles per day over the course of five days. If production is increased to 24 bicycles per day, how many additional days will it take for average production to reach 19 bicycles per day? 2. A small business had an average weekly revenue of $14,000 over the past
1. Compute the variance of the probability distribution in Table 5.2. Compute the variance of the probability distribution in Table 6.
Suppose that a pair of dice is rolled 720 times. Find the mean and standard deviation for the number of times the sum of seven appears.
A manufacturer produces smart thermostats that are packaged in boxes of 200. The probability of a thermostat being defective is .015. Find the mean and standard deviation for the number of defective thermostats in a box.
A basketball player makes each free throw with probability 3/5. Find the mean and standard deviation for the number of successes in 20 tries.
Suppose that a probability distribution has mean 35 and standard deviation 5. Use the Chebychev inequality to estimate the probability that an outcome will lie from a. 25 to 45 b. 20 to 50 c. 29 to 41
Suppose that a probability distribution has mean 8 and standard deviation .4. Use the Chebychev inequality to estimate the probability that an outcome will lie from (a) 6 to 10. (b) 7.2 to 8.8. (c) 7.5 to 8.5.
For certain types of fluorescent lights, the number of hours a bulb will burn before requiring replacement has a mean of 3000 hours and a standard deviation of 250 hours. Suppose that 5000 of such bulbs are installed in an office building. Use the Chebychev inequality to estimate the number that
An electronics firm determines that the number of defective circuit boards in each batch averages 15 with standard deviation 10. Suppose that 100 batches are produced. Use the Chebychev inequality to estimate the number of batches having from 0 to 30 defective circuit boards.
1. Suppose that a probability distribution has mean 75 and standard deviation 6. Use the Chebychev inequality to find the value of c for which the probability that the outcome lies between 75 - c and 75 + c, inclusive, is at least 7/16. 2. Suppose that a probability distribution has mean 17 and
The probability distribution for the sum of numbers obtained from rolling a pair of dice is given in Table 9.(a) Compute the mean and the variance of this probability distribution.(b) Using the table, give the probability that the number is between 4 and 10, inclusive.(c) Use the Chebychev
The probability distribution, rounded to the nearest thousandth, for the number of ones obtained from rolling 12 dice is given in Table 10. This probability distribution has mean 2 and standard deviation 1.291 (σ2=5/3). (a) Using the table, give the probability that the number of ones rolled is
Use the alternative formula for variance to calculate the variance of the random variable in Table 11.
If X is a random variable, then the variance of X equals the variance of X - a for any number a. Redo below Exercise, using this result with a = 70.
If X is a random variable, then the variance of aX equals a2 times the variance of X. Verify this result for the random variable in Exercise 2 with a = 2.
If X is a random variable, then E(X - a) = E(X ) - a and E(aX ) = aE(X ) for any number a. Give intuitive justifications of these results
Table 12 gives the fall 2015 enrollments of 10 of the largest university campuses in the United States. Determine the population mean and standard deviation for these enrollments. Which schools have enrollments within one standard deviation of the mean?
Table 13 gives the tuition, fees, room, and board for eight of the priciest colleges in the United States for the academic year 2015-16. Determine the population mean and standard deviation of these costs. Which schools have costs at least one standard deviation greater than the mean?
Table 14 summarizes the number of Ph.D. degrees awarded at two universities during the past 25 years. For example, three Ph.D. degrees were awarded at University A during 5 of the last 25 years.(a) Find the population mean and standard deviation for the number of degrees awarded each year at each
Table 15 gives the probability distributions for the possible earnings from two games of chance.(a) Find the mean and standard deviation for the earnings in each game. (b) Which game is more favorable for a gambler in the long run? (c) Which game produces more consistent results?
1. Determine by inspection which one of the probability distributions, A or B, in Fig. 5 has the greater variance.2. Determine by inspection which one of the probability distributions, C or D, in Fig. 6 has the greater variance.
Table 7 gives the probability distribution for the possible returns from two different investments.(a) Compute the mean and the variance for each investment.(b) Which investment has the higher expected return (i.e., mean)?(c) Which investment is less risky (i.e., has lesser variance)?
Two golfers recorded their scores for 20 nine-hole rounds of golf. Golfer A's scores were 39, 39, 40, 40, 40, 40, 40, 40, 41, 41, 41, 41, 41, 41, 41, 42, 43, 43, 43, 44 Golfer B's scores were 40, 40, 40, 41, 41, 41, 41, 42, 42, 42, 42, 42, 43, 43, 43, 43, 43, 43, 44, 44 (a) Compute the sample mean
Table 8 gives the relative frequency distribution for the weekly sales of two businesses.(a) Compute the population mean and the variance for each business. (b) Which business has the better sales record? (c) Which business has the more consistent sales record?
Student A received the following course grades during her first year of college: 4, 4, 4, 3, 3, 3, 2, 2, 2, 1 Student B received the following course grades during her first year: 4, 4, 4, 4, 4, 4, 3, 1, 1, 1 (a) Compute the population means and variances. (b) Which student had the better grade
Suppose that a coin is tossed 10 times. Find the mean and standard deviation for the number of heads.
In below Exercises, use the table for A(z) (Table 2) to find the areas of the shaded regions under the standard normal curve.1.2. 3. 4.
In below Exercises, find the value of z for which the area of the shaded region under the standard normal curve is as specified.1. Area is .5468.2. Area is .6915.
1. What is the 80th percentile of the standard normal distribution? 2. What is the 40th percentile of the standard normal distribution?
In below Exercises, determine µ and Ïƒ by inspection.1.2. 3. 4.
Following Exercises refer to the normal curve with µ = 8, σ = 3/4. 1. Convert 4 into standard deviations from the mean. 2. Convert 9 1/4 into standard deviation from the mean. 3. What value is exactly 10 standard deviations above the mean? 4. What value is exactly 2 standard deviations below the
In below Exercises, use the table for A(z) (Table 2) to find the areas of the shaded regions under the standard normal curve.1.2. 3. 4.
In below Exercises, find the areas of the shaded regions under the given normal curves.1.2.
In below Exercises, find the areas of the shaded regions under the given normal curves.1.2.
1. What is the probability that an outcome of a normal random variable is within two standard deviations of the mean? 2. What is the probability that an outcome of a normal random variable is within 2.5 standard deviations of the mean?
1. Find the value of s for a normal random variable X having µ = 5, if Pr (X ≤ 6) = .9772 2. Find the value of s for a normal random variable X having µ = 10, if Pr (14.5 ≤ X) = .0013.
1. Suppose that the height (at the shoulder) of adult bull African bush elephants is normally distributed with µ = 3.3 meters and σ = .2 meter. The elephant on display at the Smithsonian Institution has height 4 meters and is the largest elephant on record. What is the probability that an adult
1. Bolts produced by a machine are acceptable provided that their length is within the range from 5.95 to 6.05 centimeters Suppose that the lengths of the bolts produced are normally distributed with µ = 6 centimeters and σ = .04. What is the probability that a bolt will be of an acceptable
1. In a certain manufacturing process, lengths (in cm) of bolts are normally distributed with µ = 5.4 and σ = .6. Find the probability that a bolt selected at random has a length greater than 5.832 cm. 2. As measured with the Stanford-Binet Intelligence Scale, IQ scores are normally distributed
1. The number of barrels of oil produced yearly by a specific oil well is normally distributed with µ = 7500 barrels and σ = 1000. If the well owner has a yearly quota of 9750 barrels, what is the probability that the well's production will meet or exceed this quota? 2. Suppose that the lifetimes
1. Assume that SAT verbal scores for a first-year class at a university are normally distributed with mean 520 and standard deviation 75. (a) The top 10% of the students are placed into the honors program for English. What is the lowest score for admittance into the honors program? (b) What is the
1. A mail-order house uses an average of 300 mailing bags per day. The number of bags needed each day is approximately normally distributed with σ = 50. How many bags must the company have on hand at the beginning of a day to be 95% certain that all orders can be filled?
1. The lifetime of a certain brand of tires is normally distributed with mean m = 30,000 miles and standard deviation µ = 5000 miles. The company has decided to issue a warranty for the tires but does not want to replace more than 2% of the tires that it sells. At what mileage should the warranty
Let X be the amount of soda released by a soft-drink dispensing machine into a 6-ounce cup. Assume that X is normally distributed with σ = .25 ounces and that the average "fill" can be set by the vendor. (a) At what quantity should the average "fill" be set so that no more than .5% of the releases
1. (True or False) A normal curve with a large value of s will be flatter than one with a small value of σ. 2. Let X be a random variable with m = 4 and s = .5. (a) Use the Chebychev inequality to estimate Pr (3 ... X ... 5). (b) If X were normally distributed, what would be the exact probability
In below Exercises, find the value of z for which the area of the shaded region under the standard normal curve is as specified.1. Area is .0401.2. Area is .0456.
In following Exercises, use the normal curve to approximate the probability. 1. An experiment consists of 25 binomial trials, each having probability 1/5 of success. Use an approximating normal curve to estimate the probability of (a) exactly 5 successes. (b) between 3 and 7 successes,
An advertising agency, which reached 25% of its target audience with its old campaign, has devised a new advertising campaign. In a sample of 1000 people, it finds that 290 people have been reached by the new advertising campaign. What is the probability that at least 290 people would have been
A washing machine manufacturer finds that 2% of its washing machines break down within the first year. Find the probability that less than 15 out of a lot of 1000 washers break down within 1 year.
The incidence of color blindness among the men in a certain country is 20%. Find the expected number of color-blind men in a random sample of 70 men. What is the probability of finding exactly that number of color-blind men in a sample of size 70?
The probabilities of failure for each of three independent components in a device are .01, .02, and .01, respectively. The device fails only if all three components fail. Out of a lot of 1 million devices, how many would be expected to fail? Find the probability that more than three devices in the
In a random sample of 250 college students, 175 of them own a smartphone. Estimate the probability that a college student chosen at random owns a smartphone. If actually 75% of all college students own a smartphone, what is the probability that, in a random survey of 250 students, at most 175 of
A marksman hits a target with probability .35. Estimate the probability of hitting the target from 30 to 40 times in 100 attempts.
An airline accepts 150 reservations for a flight on an airplane that holds 140 passengers. If the Probability of a passenger for this flight cancelling is.14, estimate the probability that some passengers will have to be bumped.
A travel agent is arranging a tour for the local 1000-member ski club. They need a minimum of 29 people to register and think that the probability of a member registering for the tour is .03. Estimate the probability that enough members will register.
A fair coin is tossed 100 times. Estimate the probability that more than 65 heads or more than 65 tails appear.
In 100 tosses of a fair coin, let X be the number of heads. Estimate Pr (49 ≤ X ≤ 51).
In following Exercises, use the normal curve to approximate the probability. An experiment consists of 18 binomial trials, each having probability 2/3 of success. Use an approximating normal curve to estimate the probability of (a) exactly 10 successes. (b) between 8 and 16 successes,
Let X is the number of 4s in 120 rolls of a fair die. Estimate Pr (17 ≤ X ≤ 21).
Say that there is a 20% chance that a person chosen at random from the population has never heard of John Steinbeck. Estimate the probability that, in 150 people, we find exactly 30 people who have not heard of Steinbeck.
About 5% of American males are 6 feet 2 inches tall or taller. Estimate the probability that, of 150 men at a business meeting, no more than five are 6 feet 2 inches tall or taller.
Laboratory mice are given an illness for which the usual recovery rate is 1/6. A new drug is tested on 20 of the mice, and 8 of them recover. What is the probability that 8 or more would have recovered if the 20 mice had not been given the drug?
person claims to have ESP (extrasensory perception). A coin is tossed 16 times, and each time, the person is asked to predict in advance whether the coin will land heads or tails. The person predicts correctly 75% of the time (i.e., on 12 tosses). What is the probability of being correct 12 or more
A wine-taster claims that she can usually distinguish between domestic and imported wines. As a test, she is given 100 wines to test and correctly identifies 63 of them. What is the probability that she accomplished that good of a record by pure guessing? That is, what is the probability? of being
In American roulette, the probability of winning when betting "red" is 9/19. What is the probability of being ahead after betting the same amount 90 times?
A basketball player makes each free throw with probability 3/4. What is the probability of making 68 or more shots out of 75 trials?
A bookstore determines that 2/5 of the people who come into the store make a purchase. What is the probability that of the 54 people who come into the store during a certain hour, less than 14 make a purchase?
A baseball player gets a hit with probability .310. Find the probability that they get at least 6 hits in 20 times at bat.
1. Do you think that the probability that both balls are red is higher if the first ball is replaced before the second ball is drawn or if the first ball is not replaced? 2. Suppose that the first ball is replaced before the second ball is drawn. Find the probability that both balls are red. An urn
1. Suppose that the first ball is not replaced before the second ball is drawn. Find the probability that both balls are red. 2. Was your intuitive guess in part 1 correct? 3. Do you think that the expected number of red balls drawn is higher if the balls are drawn with replacement or without
1. Suppose that the first ball is replaced before the second ball is drawn. Find the expected number of red balls that will be drawn. 2. Was your intuitive guess in part 5 correct? 3. Suppose that the first ball is not replaced before the second ball is drawn. Find the expected number of red balls
Pretend that the 10 balls are ping-pong balls, that they have been finely ground up, and that the red and white specks have been thoroughly mixed. Forty percent of the specks will be red, and 60% will be white. Suppose that you stir the specks and use a tablespoon to scoop out 10% of the specks.
What is a bar chart? A pie chart? A histogram? A box plot?
1. What does the Chebychev inequality do? 2. What is meant by a normal random variable? 3. What is meant by the pth percentile of a normal random variable?
How are binomial probabilities approximated with the normal distribution?
What is the median of a list of numbers? The first quartile? The third quartile? The inter quartile range? The five-number summary?
What is a frequency distribution? A relative frequency distribution? A probability distribution?
How is a histogram constructed from a distribution?
1. What is a random variable? 2. What is meant by the probability distribution of a random variable?
1. What are the identifying features of a binomial random variable? 2. What is the formula for the probability of k successes in n independent binomial trials?
What is intuitively meant by the expectation (or expected value) of a random variable? Variance? Standard deviation?
Display the data from Table 1 in a bar chart that shows frequencies on the y-axis. Then display the data in a pie chart
The probability distribution of a random variable X is given in the table. Determine the mean, variance, and standard deviation of X.
Lucy and Ethel play a game of chance in which a pair of fair dice is rolled once. If the result is 7 or 11, then Lucy pays Ethel $10. Otherwise, Ethel pays Lucy $3. In the long run, which player comes out ahead, and by how much?
1. Suppose that a probability distribution has mean 10 and standard deviation 1/3. Use the Chebychev inequality to estimate the probability that an outcome will lie between 9 and 11. 2. Suppose that a probability distribution has mean 50 and standard deviation 8. Use the Chebychev inequality to
1. Find the area of the shaded region under the normal curve with µ = 5, Ïƒ = 3 shown in Fig. 2(a).2. Find the area of the shaded region under the standard normal curve shown in Fig. 2(b).
1. The height of adult males in the United States is normally distributed with µ = 5.75 feet andÏƒ = .2 feet. What percent of the adult male population has height of 6 feet or greater?2. Figure 3(a) is a standard normal curve. Find the value of z for which the area of the shaded
Figure 3(b) is a normal curve with µ = 80 and σ = 15. Find the value of h for which the area of the shaded region is .8664.
Find the five-number summary and the inter quartile range for the following set of numbers, and then draw the box plot: 1, 2, 3, 4, 5, 9, 14, 23
As measured with the Wechsler Adult Intelligence Scale, IQ scores are normally distributed with mean 100 and standard deviation 15. (a) What percent of the adult population has an IQ score of 133 or more? (b) Find the 95th percentile of IQ scores.
In a certain city, 2/5 of the registered voters are women. Out of a group of 54 voters allegedly selected at random for jury duty, 13 are women. A local civil liberties group has charged that the selection procedure discriminated against women. Use the normal curve to estimate the probability of 13
In a complicated production process, 1/4 of the items produced have to be readjusted. Use the normal curve to estimate the probability that out of a batch of 75 items, between 8 and 22 (inclusive) of the items require readjustment.
Give an example of a grade distribution for a class of students in which (a) Scoring in the 3rd quartile is not very good. (b) Scoring in the 3rd quartile corresponds to a perfect grade.
Give an example of a distribution of 10 grades for which (a) The mean and median are equal. (b) The mean is less than the median. (c) The median is less than the mean.
What is the difference between a population mean and a sample mean?
1. Explain in your own words the meaning of sample mean. 2. If each number in a set of numbers is increased by 5, will the mean increase by 5?
1. If each number in a set of numbers is doubled, will the standard deviation be doubled? 2. Explain the type of probability situations for which the binomial distribution applies. 3. Give an example of a sequence of repeated trials that does not produce a binomial distribution.

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