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descriptive statistics
Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying Ye - Solutions
A maker of a certain brand of low-fat cereal bars claims that the average saturated fat content is 0.5 gram. In a random sample of 8 cereal bars of this brand, the saturated fat content was 0.6, 0.7, 0.7, 0.3, 0.4, 0.5, 0.4, and 0.2. Would you agree with the claim?Assume a normal distribution.
A normal population with unknown variance has a mean of 20. Is one likely to obtain a random sample of size 9 from this population with a mean of 24 and a standard deviation of 4.1? If not, what conclusion would you draw?
A manufacturing firm claims that the batteries used in their electronic games will last an average of 30 hours. To maintain this average, 16 batteries are tested each month. If the computed t-value falls between −t0.025 and t0.025, the firm is satisfied with its claim. What conclusion should the
Given a random sample of size 24 from a normal distribution, find k such that(a) P(−2.069
(a) Find P(−t0.005
(a) Find P(T < 2.365) when v = 7.(b) Find P(T > 1.318) when v = 24.(c) Find P(−1.356 −2.567) when v = 17.
(a) Find t0.025 when v = 14.(b) Find −t0.10 when v = 10.(c) Find t0.995 when v = 7.
Show that the variance of S2 for random samples of size n from a normal population decreases as n becomes large. [Hint: First find the variance of(n − 1)S2/σ2.]
The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean μ = 74 and a varianceσ2 = 8. Would you still consider σ2 = 8 to be a valid value of the variance if a random sample of 20 students who take the placement test this
Assume the sample variances to be continuous measurements. Find the probability that a random sample of 25 observations, from a normal population with variance σ2 = 6, will have a sample variance S2(a) greater than 9.1;(b) between 3.462 and 10.745.
For a chi-squared distribution, find χ2α such that(a) P(X2 > χ2α)=0.01 when v = 21;(b) P(X2 < χ2α)=0.95 when v = 6;(c) P(χ2α < X2 < 23.209) = 0.015 when v = 10.
For a chi-squared distribution, find χ2α such that(a) P(X2 > χ2α)=0.99 when v = 4;(b) P(X2 > χ2α)=0.025 when v = 19;(c) P(37.652 < X2 < χ2α)=0.045 when v = 25.
For a chi-squared distribution, find(a) χ2 0.005 when v = 5;(b) χ2 0.05 when v = 19;(c) χ2 0.01 when v = 12.
For a chi-squared distribution, find(a) χ2 0.025 when v = 15;(b) χ2 0.01 when v = 7;(c) χ2 0.05 when v = 24.
Let X1, X2,...,Xn be a random sample from a distribution that can take on only positive values. Use the Central Limit Theorem to produce an argument that if n is sufficiently large, then Y = X1X2 ··· Xn has approximately a lognormal distribution
Consider the situation described in Example 8.4 on page 234. Do these results prompt you to question the premise that μ = 800 hours? Give a probabilistic result that indicates how rare an event X¯ ≤ 775 is when μ = 800. On the other hand, how rare would it be if μ truly were, say, 760 hours?
Two alloys A and B are being used to manufacture a certain steel product. An experiment needs to be designed to compare the two in terms of maximum load capacity in tons (the maximum weight that can be tolerated without breaking). It is known that the two standard deviations in load capacity are
The chemical benzene is highly toxic to humans. However, it is used in the manufacture of many medicine dyes, leather, and coverings. Government regulations dictate that for any production process involving benzene, the water in the output of the process must not exceed 7950 parts per million (ppm)
Two different box-filling machines are used to fill cereal boxes on an assembly line. The critical measurement influenced by these machines is the weight of the product in the boxes. Engineers are quite certain that the variance of the weight of product is σ2 = 1 ounce.Experiments are conducted
Consider Case Study 8.2 on page 238. Suppose 18 specimens were used for each type of paint in an experiment and ¯xA −x¯B, the actual difference in mean drying time, turned out to be 1.0.(a) Does this seem to be a reasonable result if the two population mean drying times truly are equal?Make use
The mean score for freshmen on an aptitude test at a certain college is 540, with a standard deviation of 50. Assume the means to be measured to any degree of accuracy. What is the probability that two groups selected at random, consisting of 32 and 50 students, respectively, will differ in their
The distribution of heights of a certain breed of terrier has a mean of 72 centimeters and a standard deviation of 10 centimeters, whereas the distribution of heights of a certain breed of poodle has a mean of 28 centimeters with a standard deviation of 5 centimeters.Assuming that the sample means
A random sample of size 25 is taken from a normal population having a mean of 80 and a standard deviation of 5. A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard deviation of 3. Find the probability that the sample mean computed from
In a chemical process, the amount of a certain type of impurity in the output is difficult to control and is thus a random variable. Speculation is that the population mean amount of the impurity is 0.20 gram per gram of output. It is known that the standard deviation is 0.1 gram per gram. An
The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean μ = 3.2 minutes and a standard deviationσ = 1.6 minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller’s window is(a) at most 2.7
The average life of a bread-making machine is 7 years, with a standard deviation of 1 year. Assuming that the lives of these machines follow approximately a normal distribution, find(a) the probability that the mean life of a random sample of 9 such machines falls between 6.4 and 7.2 years;(b) the
If a certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms, what is the probability that a random sample of 36 of these resistors will have a combined resistance of more than 1458 ohms?
The random variable X, representing the number of cherries in a cherry puff, has the following probability distribution:x 4567 P(X = x) 0.2 0.4 0.3 0.1(a) Find the mean μ and the variance σ2 of X.(b) Find the mean μX¯ and the variance σ2 X¯ of the mean X¯ for random samples of 36 cherry
The heights of 1000 students are approximately normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. Suppose 200 random samples of size 25 are drawn from this population and the means recorded to the nearest tenth of a centimeter. Determine(a) the mean
A soft-drink machine is regulated so that the amount of drink dispensed averages 240 milliliters with a standard deviation of 15 milliliters. Periodically, the machine is checked by taking a sample of 40 drinks and computing the average content. If the mean of the 40 drinks is a value within the
Given the discrete uniform population f(x) = 1 3 , x = 2, 4, 6, 0, elsewhere, find the probability that a random sample of size 54, selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4. Assume the means are measured to the nearest tenth.
A certain type of thread is manufactured with a mean tensile strength of 78.3 kilograms and a standard deviation of 5.6 kilograms. How is the variance of the sample mean changed when the sample size is(a) increased from 64 to 196?(b) decreased from 784 to 49?
If the standard deviation of the mean for the sampling distribution of random samples of size 36 from a large or infinite population is 2, how large must the sample size become if the standard deviation is to be reduced to 1.2?
If all possible samples of size 16 are drawn from a normal population with mean equal to 50 and standard deviation equal to 5, what is the probability that a sample mean X¯ will fall in the interval from μX¯ −1.9σX¯to μX¯ −0.4σX¯ ? Assume that the sample means can be measured to any
In the 2004-05 football season, University of Southern California had the following score differences for the 13 games it played.11 49 32 3 6 38 38 30 8 40 31 5 36 Find(a) the mean score difference;(b) the median score difference.
Verify that the variance of the sample 4, 9, 3, 6, 4, and 7 is 5.1, and using this fact, along with the results of Exercise 8.14, find(a) the variance of the sample 12, 27, 9, 18, 12, and 21;(b) the variance of the sample 9, 14, 8, 11, 9, and 12.
(a) Show that the sample variance is unchanged if a constant c is added to or subtracted from each value in the sample.(b) Show that the sample variance becomes c2 times its original value if each observation in the sample is multiplied by c.
The grade-point averages of 20 college seniors selected at random from a graduating class are as follows:Calculate the standard deviation.
The tar contents of 8 brands of cigarettes selected at random from the latest list released by the Federal Trade Commission are as follows: 7.3, 8.6, 10.4, 16.1, 12.2, 15.1, 14.5, and 9.3 milligrams. Calculate(a) the mean;(b) the variance.
For the data of Exercise 8.5, calculate the variance using the formula(a) of form (8.2.1);(b) in Theorem 8.1.
For the sample of reaction times in Exercise 8.3, calculate(a) the range;(b) the variance, using the formula of form (8.2.1).
Consider the data in Exercise 8.2, find(a) the range;(b) the standard deviation.
According to ecology writer Jacqueline Killeen, phosphates contained in household detergents pass right through our sewer systems, causing lakes to turn into swamps that eventually dry up into deserts. The following data show the amount of phosphates per load
A random sample of employees from a local manufacturing plant pledged the following donations, in dollars, to the United Fund: 100, 40, 75, 15, 20, 100, 75, 50, 30, 10, 55, 75, 25, 50, 90, 80, 15, 25, 45, and 100. Calculate(a) the mean;(b) the mode.
Find the mean, median, and mode for the sample whose observations, 15, 7, 8, 95, 19, 12, 8, 22, and 14, represent the number of sick days claimed on 9 federal income tax returns. Which value appears to be the best measure of the center of these data? State reasons for your preference.
The numbers of incorrect answers on a true-false competency test for a random sample of 15 students were recorded as follows: 2, 1, 3, 0, 1, 3, 6, 0, 3, 3, 5, 2, 1, 4, and 2. Find(a) the mean;(b) the median;(c) the mode.
The number of tickets issued for traffic violations by 8 state troopers during the Memorial Day weekend are 5, 4, 7, 7, 6, 3, 8, and 6.(a) If these values represent the number of tickets issued by a random sample of 8 state troopers from Montgomery County in Virginia, define a suitable
The reaction times for a random sample of 9 subjects to a stimulant were recorded as 2.5, 3.6, 3.1, 4.3, 2.9. 2.3, 2.6, 4.1, and 3.4 seconds. Calculate(a) the mean;(b) the median.
The lengths of time, in minutes, that 10 patients waited in a doctor’s office before receiving treatment were recorded as follows: 5, 11, 9, 5, 10, 15, 6, 10, 5, and 10. Treating the data as a random sample, find(a) the mean;(b) the median;(c) the mode.
Define suitable populations from which the following samples are selected:(a) Persons in 200 homes in the city of Richmond are called on the phone and asked to name the candidate they favor for election to the school board.(b) A coin is tossed 100 times and 34 tails are recorded.(c) Two hundred
By expanding etx in a Maclaurin series and integrating term by term, show that MX(t) = ∞−∞
If both X and Y , distributed independently, follow exponential distributions with mean parameter 1, find the distributions of(a) U = X + Y ;(b) V = X/(X + Y ).
Using the moment-generating function of Exercise 7.21, show that the mean and variance of the chisquared distribution with v degrees of freedom are, respectively, v and 2v.
Show that the moment-generating function of the random variable X having a chi-squared distribution with v degrees of freedom is MX(t) = (1 − 2t)−v/2.
The moment-generating function of a certain Poisson random variable X is given by MX(t) = e 4(et−1).Find P(μ − 2σ
A random variable X has the Poisson distribution p(x; μ) = e−μμx/x! for x = 0, 1, 2,... . Show that the moment-generating function of X is MX(t) = eμ(et−1).Using MX(t), find the mean and variance of the Poisson distribution.
A random variable X has the geometric distribution g(x; p) = pqx−1 for x = 1, 2, 3,... . Show that the moment-generating function of X is MX(t) = pet 1 − qet , t< ln q, and then use MX(t) to find the mean and variance of the geometric distribution.
A random variable X has the discrete uniform distribution f(x; k) = 1 k , x = 1, 2,..., k, 0, elsewhere.Show that the moment-generating function of X is MX(t) = et(1 − ekt)k(1 − et) .
Show that the rth moment about the origin of the gamma distribution isμr = βrΓ(α + r)Γ(α) .[Hint: Substitute y = x/β in the integral defining μr and then use the gamma function to evaluate the integral.]
Let X have the probability distribution f(x) = 2(x+1)9 , −1
Let X be a random variable with probability distribution f(x) = 1+x 2 , −1
A current of I amperes flowing through a resistance of R ohms varies according to the probability distribution f(i) = 6i(1 − i), 0
Let X1 and X2 be independent random variables each having the probability distribution f(x) = e−x, x> 0, 0, elsewhere.Show that the random variables Y1 and Y2 are independent when Y1 = X1 + X2 and Y2 = X1/(X1 + X2).
The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Assume that the joint density function of these variables is given by f(x, y) = 2, 0 < x < y, 0
The random variables X and Y , representing the weights of creams and toffees, respectively, in 1-kilogram boxes of chocolates containing a mixture of creams, toffees, and cordials, have the joint density function f(x, y) = 24xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x + y ≤ 1, 0, elsewhere.(a) Find the
The hospital period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X + 4, where X has the density function f(x) = 32(x+4)3 , x> 0, 0, elsewhere.(a) Find the probability density function of the random variable Y .(b) Using the density
A dealer’s profit, in units of $5000, on a new automobile is given by Y = X2, where X is a random variable having the density function f(x) = 2(1 − x), 0
The speed of a molecule in a uniform gas at equilibrium is a random variable V whose probability distribution is given by f(v) = kv2e−bv2, v> 0, 0, elsewhere, where k is an appropriate constant and b depends on the absolute temperature and mass of the molecule.Find the probability distribution
Given the random variable X with probability distribution f(x) = 2x, 0
Let X have the probability distribution f(x) = 1, 0
Let X1 and X2 be discrete random variables with joint probability distribution f(x1, x2) = x1x2 18 , x1 = 1, 2; x2 = 1, 2, 3, 0, elsewhere.Find the probability distribution of the random variable Y = X1X2.
Let X1 and X2 be discrete random variables with the joint multinomial distribution f(x1, x2)=for x1 = 0, 1, 2; x2 = 0, 1, 2; x1 + x2 ≤ 2; and zero elsewhere. Find the joint probability distribution of Y1 = X1 + X2 and Y2 = X1 − X2.
Let X be a binomial random variable with probability distribution f(x) = 3 x 2 5x 3 53−x , x = 0, 1, 2, 3, 0, elsewhere.Find the probability distribution of the random variable Y = X2.
Let X be a random variable with probability f(x) = 1 3 , x = 1, 2, 3, 0, elsewhere.Find the probability distribution of the random variable Y = 2X − 1.
Group Project: Have groups of students observe the number of people who enter a specific coffee shop or fast food restaurant over the course of an hour, beginning at the same time every day, for two weeks.The hour should be a time of peak traffic at the shop or restaurant. The data collected will
The length of time, in seconds, that a computer user takes to read his or her e-mail is distributed as a lognormal random variable with μ = 1.8 and σ2 = 4.0.(a) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute?(b) What is the probability that a user
From the relationship between the chi-squared random variable and the gamma random variable, prove that the mean of the chi-squared random variable is v and the variance is 2v.
Explain why the nature of the scenario in Review Exercise 6.82 would likely not lend itself to the exponential distribution.
Derive the cdf for the Weibull distribution.[Hint: In the definition of a cdf, make the transformation z = yβ.]
The length of life, in hours, of a drill bit in a mechanical operation has a Weibull distribution withα = 2 and β = 50. Find the probability that the bit will fail before 10 hours of usage.
The length of time between breakdowns of an essential piece of equipment is important in the decision of the use of auxiliary equipment. An engineer thinks that the best model for time between breakdowns of a generator is the exponential distribution with a mean of 15 days.(a) If the generator has
In a human factor experimental project, it has been determined that the reaction time of a pilot to a visual stimulus is normally distributed with a mean of 1/2 second and standard deviation of 2/5 second.(a) What is the probability that a reaction from the pilot takes more than 0.3 second?(b) What
Consider Review Exercise 6.78. Given the assumption of the exponential distribution, what is the mean number of calls per hour? What is the variance in the number of calls per hour?
Consider now Review Exercise 3.74 on page 108.The density function of the time Z in minutes between calls to an electrical supply store is given by f(z) = 1 10 e−z/10, 0
The beta distribution has considerable application in reliability problems in which the basic random variable is a proportion, as in the practical scenario illustrated in Exercise 6.50 on page 206. In that regard, consider Review Exercise 3.73 on page 108. Impurities in batches of product of a
In Exercise 6.54 on page 206, the lifetime of a transistor is assumed to have a gamma distribution with mean 10 weeks and standard deviation √50 weeks.Suppose that the gamma distribution assumption is incorrect. Assume that the distribution is normal.(a) What is the probability that a transistor
For Review Exercise 6.74, what is the mean of the average water usage per hour in thousands of gallons?
The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution with parametersμ = 5 and σ = 2. It is important for planning purposes to get a sense of periods of high usage. What is the probability that, for any given hour,
For Review Exercise 6.72, what are the mean and variance of the time that elapses before 2 failures occur?
Consider the information in Review Exercise 6.66. What is the probability that less than 200 hours will elapse before 2 failures occur?
A technician plans to test a certain type of resin developed in the laboratory to determine the nature of the time required before bonding takes place. It is known that the mean time to bonding is 3 hours and the standard deviation is 0.5 hour. It will be considered an undesirable product if the
A controlled satellite is known to have an error(distance from target) that is normally distributed with mean zero and standard deviation 4 feet. The manufacturer of the satellite defines a success as a firing in which the satellite comes within 10 feet of the target.Compute the probability that
The elongation of a steel bar under a particular load has been established to be normally distributed with a mean of 0.05 inch and σ = 0.01 inch. Find the probability that the elongation is(a) above 0.1 inch;(b) below 0.04 inch;(c) between 0.025 and 0.065 inch.
For an electrical component with a failure rate of once every 5 hours, it is important to consider the time that it takes for 2 components to fail.(a) Assuming that the gamma distribution applies, what is the mean time that it takes for 2 components to fail?(b) What is the probability that 12 hours
In a chemical processing plant, it is important that the yield of a certain type of batch product stay above 80%. If it stays below 80% for an extended period of time, the company loses money. Occasional defective batches are of little concern. But if several batches per day are defective, the
A certain type of device has an advertised failure rate of 0.01 per hour. The failure rate is constant and the exponential distribution applies.(a) What is the mean time to failure?(b) What is the probability that 200 hours will pass before a failure is observed?
According to a recent census, almost 65% of all households in the United States were composed of only one or two persons. Assuming that this percentage is still valid today, what is the probability that between 590 and 625, inclusive, of the next 1000 randomly selected households in America consist
A manufacturer of a certain type of large machine wishes to buy rivets from one of two manufacturers. It is important that the breaking strength of each rivet exceed 10,000 psi. Two manufacturers (A and B) offer this type of rivet and both have rivets whose breaking strength is normally
When α is a positive integer n, the gamma distribution is also known as the Erlang distribution.Setting α = n in the gamma distribution on page 195, the Erlang distribution is f(x) = xn−1e−x/ββn(n−1)! , x> 0, 0, elsewhere.It can be shown that if the times between successive events are
The exponential distribution is frequently applied to the waiting times between successes in a Poisson process. If the number of calls received per hour by a telephone answering service is a Poisson random variable with parameter λ = 6, we know that the time, in hours, between successive calls has
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