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Questions and Answers of Statistics

Consider the 1/16 of the 27 factorial discussed in Section 15.11. List the additional 11 defining contrasts.
Construct a Plackett-Burman design for 10 variables containing 24 experimental runs.
A Plackett-Burman design is used for the purpose of studying the rheological properties of high molecular-weight, copolymers. Two levels of each of six variables are fixed in the experiment. The
A large petroleum company in the Southwest regularly conducts experiments to test additives to drilling fluids. Plastic viscosity is a rheological measure reflecting the thickness of the fluid.
A 22 factorial experiment is analyzed by the Statistics Consulting Center at Virginia Polytechnic Institute and State University. The client is a member of the Department of Housing, Interior Design,
Construct a design involving 12 runs where 2 factors are varied at 2 levels each. You are further restricted in that blocks of size 2 must be used, and you must be able to make significance tests on
In the study The Use of Regression Analysis for Correcting Matrix Effects in the X-Ray Fluorescence Analysis of Pyrotechnic Compositions, published in the Proceedings of the Tenth Conference on the
Show the blocking scheme for a 27 factorial experiment in eight blocks of size 16 each, using ABCD, CDEFG, and BDF as defining contrasts. Indicate which interactions are completely sacrificed in the
Use Table to construct a 16-run design with 8 factors that is resolutionIV.
In your design of Review Exercise 15.47, verify that the design is indeed resolution IV.
Construct a design that contains nine design points, is orthogonal, contains 12 total runs, 3 degrees of freedom for replication error, and allows for a lack of fit test for pure quadratic curvature.
Consider a design which is a 23-1 w4th 2 center runs. Consider y f as the average response at the design parameter and j/o as the average response at the design center. Suppose the true regression
Define suitable populations from which the following samples are selected:(a) Persons in 200 homes are called by telephone in the city of Richmond and asked to name the candidate that they favor for
The number of tickets issued for traffic violations by 8 state troopers during the Memorial Day weekend is 5, 4, 7, 7, 6, 3, 8, and 6.(a) If these values represent the number of tickets issued by a
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
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
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.
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
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,
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
With reference to the lengths of time that 10 patients waited in a doctor's office before receiving treatment in Exercise 8.4, find(a) The range;(b) The standard deviation.
With reference to the sample of reaction times for the 9 subjects receiving the stimulant in Exercise 8.5, calculate (a) The range;(b) The variance using the formula of Definition 8.6.
With reference to the random sample of incorrect answers on a true-false competency test for the 15 students in Exercise 8.3, calculate the variance using the formula(a) Of Definition 8.6;(b) Of
The tar contents of 8 brands of cigarettes selected at random from the latest list released by the federal Trade Commission is as follows: 7.3, 8.6, 10.4, 16.1, 12.2, 15.1, 14.5, and 9.3 milligrams.
The grade-point averages of 20 college seniors selected at random from a graduating class are as follows: Calculate the standarddeviation.
(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
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
In the season of 2004-05, the football team of University of South California had the following score differences for its 13 games played. 11 49
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
Given the discrete uniform population 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 to
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
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 size of the sample become if the standard
A soft-drink machine is being 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
The heights of 1000 students are approximately normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. If 200 random samples of size 25 are drawn from this
The random variable X, representing the number of cherries in a cherry puff, has the following probability distribution:(a) Find the mean p. and the variance ?2 of X.(b) Find the mean px, and 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
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
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 deviation σ = 1.6 minutes. If a random sample of 64
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 means amount of the impurity is
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
The distribution of heights of a certain breed of terrier dogs has a mean height of 72 centimeters and a standard deviation of 10 centimeters, whereas the distribution of heights of a certain breed
The mean score for freshmen on an aptitude test at a certain college is 540, with a standard deviation of 50. What is the probability that two groups of students selected at random, consisting of 32
Construct a quantile plot of these data. The lifetimes, in hours, of fifty 40-watt, 110-volt internally frosted incandescent lamps taken from forced lifetests:
Consider Example 8.8 Suppose 18 specimens were used for each type of paint in an experiment and xA — xB, the actual difference in mean drying time turned out to be 1.0. (a) Does this seem to
Two different box-filling machines are used to fill cereal boxes on the assembly line. The critical measurement influenced by these machines is the weight of the product in the machines. Engineers
Construct a normal quantile-quantile plot of these data. Diameters of 36 rivet heads in 1/100 of aninch:
The chemical benzene is highly toxic to humans. However, it is used in the manufacture of many medicine dyes, leather, and many coverings. In any production process involving benzene, the water in
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. This is the maximum that can
Consider the situation in example 8.6 do these results prompt you to question the premise that p = 800 hours? Give a probabilistic result that indicates how rare an event that X < 775 is when = 800.
Let X1, X2,..., X 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--
For a chi-squared distribution find(a) X20.025 when v = 15.(b) X20.01 when v = 7.(c) X20.05 when v = 24.
For a chi-squared distribution find the following:(a) X20.05 when v = 5.(b) X20.01 when v = 19.(c) X20.0i when v = 12.
For a chi-squared distribution find X2α .such that(a) P(X2 > X2α) = 0.99 when v = 4;(b) P(X2 > X2α) = 0.025 when v = 19:(c) P(37.652 < X2 < X2α) = 0.045 when v = 25.
For a chi-squared distribution find X2a such that(a) F(X2 > X2a) = 0.01 when v = 21:(b) P(X2 < x2a) = 0.95 when v = 6;(c) P(X2a < X2 < 23.209) = 0.015 when v = 10.
Find the probability that a random sample of 25 observations, from a normal population with variance a" = 6, will have a variance s(a) Greater than! I;(b) Between 3.462 and 10.745. Assume the sample
The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean μ = 71 and a variance a2 = 8. Would you still consider σ2
Show that the variance of S for random samples of size n. from a normal population decreases as n becomes largo.
(a) Find t0.025 when v = 14.(b) Find — t0.10 when v = 10.(c) Find t0.995 when v = 7.
(a) Find P (T < 2.3G5) when v = 7.(b) Find P (T > 1.318) when v = 24.(c) Find F (—1.356 < T < 2.179) when v = 12.(d) Find P (T > – 2.567) when r = 17.
(a) Find P(– t0.005 < T < t0.01) for v = 20.(b) Find P(T> -t0.025).
Given a random sample of size 24 from a normal distribution, find k such that(a) P (–2.069 < T < k) = 0.965;(b) P {k < T < 2.807) = 0.095;(c) P (– k < T < k) = 0.90.
A manufacturing firm claims that the batteries used in their electronic games will last an average of 30 hours. To maintain this average L6 batteries are tested each month. If the computed /-value
A normal population with unknown variance has a mean of 20. Is one likely to obtain a random sample of size from this population with a mean of 24 and a standard deviation of 4.1? If not, what
A maker of a certain brand of low-fat cereal bars claims that their 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.
For an F-distribution find(a) f0.05 with v1 = 7 and v2 = 15;(b) f0.05 with v1 = 15 and v2 = 7:(c) f0.01 with v1 = 24 and v2 = 19;(el) f0.95 with v1 = 19 and v2 = 24;(e) f0.99 with v1 = 28 and v2 = 12.
Pull-strength tests on 10 soldered leads for a semi conductor device yield the following results in pounds force required to rupture the bond: 19.8 12.7
Consider the following measurements of the heat producing capacity of the coal produced by two mines (in millions of calories per ton): Mine 1: 8260
Consider the data displayed in Exercise 1.20. Construct a box-and-whisker plot, and comment on the nature of the sample. Compute the sample mean and sample standard deviation.
If X1, X2,..., X0„ are independent random variables having identical exponential distributions with parameter 6, show that the density function of the random variable Y = X\ +X2+ • + Xn is that
In testing for carbon monoxide in a certain brand of cigarette, the data, in milligrams per cigarette, were coded by subtracting 12 from each observation. Use the results of Exercise 8.14 to find
If S21 and S22 represent the variances of independent random samples of size n1 = 8 and n2 = 12, taken from normal populations with equal variances, find the P(S21/S22 < 4.89).
A random sample of 5 bank presidents indicated annual salaries of $395,000, $521,000, $483,000, $479,000, and 8510,000. Find the variance of this set.
If the number of hurricanes that hit a certain area of the eastern United States per year is a random variable having a Poisson distribution with p = 6, find the probability that this area will be
A taxi company tests a random sample of 10 steel-belted radial tires of a certain brand and recorded the following tread wear: 48,000, 53,000. 45,000. 61,000, 59,000, 56,000, 63,000, 49,000, 53,000,
Consider the data of Exercise 1.19. Construct a box-and-whisker plot. Comment computes the sample mean and sample standard deviation.
If S21 and S22 represent the variances of independent random samples of size n1 = 25 and n2 — 31, taken from normal populations with variances σ21 = 10 and erf = 15, respectively, find
Consider Review Exercise 8.56. Comment on any outliers in the data.
The breaking strength X of a certain rivet used in a machine engine has a mean 5000 psi and standard deviation 400 psi. A random sample of 36 rivets is taken. Consider the distribution of X, the
Consider the situation of Review Exercise 8.62. If the population from which the sample was taken has population mean p = 53,000 kilometers, does the sample information here seem to support that
Two distinct solid fuel propellants, type A and type B, are being considered in a space program activity. Burning rates on the propellant are crucial. Random samples of 20 specimens of the two
The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is felt that when catalyst A is used the
From the information in Review Exercise 8.70 compute (assuming μB = 65%) P(XB > 70).
Given a normal random variable X with mean 20 and variance 9, and a random sample of size n taken from the distribution, what sample size n is necessary in order that P(19.9 < X < 20.1) = 0.95?
In Chapter 9 the concept of parameter estimation will be discussed at length. Suppose X is a random variable with mean p and variance a2 = 1.0. Suppose also that a random sample of size n is to be
Suppose a filling machine is used to fill cartons with a liquid product. The specification that is strictly enforced for the filling machine is 9 ± 1.5 oz. If any carton is produced with weight
Consider the situation in Review Exercise 8.74. Suppose a considerable quality effort is conducted to "tighten'' the variability in the system. Following the effort, a random sample of size 40 is
If A’ is a binomial random variable, show that(a) P = X/niis is a biased estimator of p.(b) P = x + √n/2/n v √2 is a biased estimator of p.
Show that the estimator P' of Exercise: 9.2 (b) becomes unbiased as n → ∞.
An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of
Many cardiac patients wear implanted pacemakers to control their heartbeat. A plastic connector module mounts on the top of the pacemaker. Assuming a standard deviation of 0.0015 and an approximate
The heights of a random sample of 50 college students showed a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters.(a) Construct a 98% confidence interval for the mean height of all
A random sample of 100 automobile owners shows that, in the state of Virginia, an automobile is driven on the average 23,500 kilometers per year with a standard deviation of 3900 kilometers. Assume
How large a sample is needed in Exercise 9.4 if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean?
How large a sample is needed in Exercise 9.5 if we wish to be 95% confident that our sample mean will be within 0.0005 inch of the true mean?
An efficiency expert wishes to determine the average time that it takes to drill three holes in a certain metal clamp. How large a sample will he need to be 95% confident that his sample mean will be
A UCLA researcher claims that the life span of mice can be extended by as much as 25% when the calories in their food are reduced by approximately 40% from the time they are weaned. The restricted
Regular consumption of presweetened cereals contributes to tooth decay, heart disease, and other degenerative diseases, according to studies conducted by Dr. W. H. Bowen of the National Institute of
A machine is producing metal pieces that are cylindrical in shape. A sample of pieces is taken and the diameters are 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimeters. Find a 99%
A random sample of 10 chocolate energy bars of a certain brand has, on average, 230 calories with a standard deviation of 15 calories. Construct a 99% confidence interval for the true mean calorie
A random sample of 12 shearing pins is taken in a study of the Rockwell hardness of the head on the pin. Measurements on the Rockwell hardness were made for each of the 12, yielding an average value

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