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statistics principles and methods

Statistics The Exploration And Analysis Of Data 6th Edition John M Scheb, Jay Devore, Roxy Peck - Solutions

Using the first two samples of Table 8.3, which have means of 0.258 and –0.491, test the null hypothesis that μ1 = μ2 against a two-tailed alternative. Assume the population variances are not known.Use α = 5%.
Three measurements of a pollutant upgradient of a landfill yield a mean of 15.4 mg/L and a standard deviation of 2.8 mg/L. Five measurements made downgradient of the landfill result in a mean of 29.1 mg/L and a standard deviation of 7.4 mg/L. Is it safe to conclude that the landfill contributes to
Destructive tests made on 8 samples of wood to determine the failure strength yield a mean of 56.4 and a standard deviation of 5.22. Nondestructive tests are made on 12 samples of the same type of wood, resulting in a mean of 53.7 and a standard deviation of 3.16. Is it reasonable to conclude that
To determine if the octane level of a gasoline influenced gas mileage, ten cars were driven over a 100-mile road test first using a 87-octane gas and then a 93-cotane gas. The miles/gallon measured for the ten cars were 87-octane: 17.6, 19.1, 18.3, 18.8, 19.5, 19.0, 18.1, 17.9, 19.2, 18.4
Two brands of resistance thermometers are available to a laboratory. To decide which brand to buy, they initially test six thermometers of each brand, with the following measured values when the known standard is 950 mv:Brand A: 938, 954, 947, 961, 951, 944 Brand B: 931, 948, 967, 973, 947, 964(a)
Wetland ecologists often recommend including an island in the middle of wetlands to increase the retention time during minor flood events. One ecologist argues that the island does not increase retention time because the travel velocity is increased along with travel length. Dye studies are used to
Office workers complain that the noise level in their office Area 1 is greater than that in the company’s other office Area 2. Six measurements are made on different days in each area, with the following results:Area 1: 61, 66, 65, 72, 59, 68 decibels Area 2: 64, 56, 53, 61, 63, 51 decibels(a)
Two concrete plants are providing concrete to a construction site. While the intended design is the same, five specimens are taken from each plant, with the following results of crushing strength:Plant 1 : 34700, 36800, 35200, 33900, 35800 kN/m2 Plant 2: 33800, 34700, 35500, 34600, 34000 kN/m2 Can
The travel times between an airport and center city is sampled for public buses and private taxicabs.A sample of eight bus trips yields times of 44, 56, 39, 42, 48, 51, 36, 43 min. Ten taxis are included in the survey with times of 37, 46, 40, 32, 35, 46, 39, 41, 29, 36 min. Given the cost
From the samples in Table 8.3, compare the sample means of the last two samples of 5 (1.412 vs.0.094). Use a two-sided alternative hypothesis and a 5% level of significance. Assume the population is unknown.
A random sample of 12 has a mean and standard deviation of 240 and 30, respectively. Test the null hypothesis that μ = 215 against the alternative hypothesis that μ > 215 at a level of significance of (a)5% and (b) 1%.
A public water supply official claims that the average household water use is greater than 350 gal/day. To test this assertion, a random sample of 200 households was questioned. If the random sample showed an average use of 359 gal/day and a standard deviation of 35 gal/day, would it be safe to
A random sample of 10 yields a mean and standard deviation, respectively, of 73.6 and 7.9. Assuming a two-sided test, test the hypothesis that the sample is from a population with a mean of 80. Use a level of significance of (a) 5% and (b) 1%.
A random sample of 25 has a mean of 4.8 with a standard deviation of 0.32. Test the null hypothesis that μ = 4.95 against the alternative hypothesis that μ < 4.95 at the 1% level of significance.
If we can compute the probabilities of type I and type II errors, what factors would contribute to the decision to select specific values of α and β?
Explain why the use of a level of significance of 5% is not best in all circumstances.
If the rejection probability of a test on a mean is 0.4%, what should be said about the decision implied about the null hypothesis?
If the rejection probability of a test on a mean is 8.4%, what should be said about the decision implied about the null hypothesis?
Define the term rejection probability. Explain its relationship with the level of significance when testing a hypothesis.
A random sample of 8 has a mean of 543. Test the null hypothesis that μ = 500 against the alternative hypothesis that μ > 120 at a 1% level of significance for the following cases; (a) assuming the population standard deviation of 45 is known, and (b) assuming that the population standard
A random sample of 10 has a mean of 110. Test the null hypothesis that μ = 120 against the alternative hypothesis that μ < 120 at the 5% level of significance for the following cases: (a) assuming that the population standard deviation of 18 is known; and (b) assuming that the population standard
A sample of 20 yields a mean of 32.4. Test the two-sided hypothesis that the sample was drawn from a population with a mean of 35 for the following cases: (a) if the variance of the population is 33; and(b) if the variance of the population is unknown, but the sample variance is 33. Use a level of
Two tests were provided for testing the hypothesis of a mean value against a standard, the Z test of Equation 9.6 and the t test of Equation 9.12. What are the differences of the two tests in terms of the purpose, data analysis requirements, underlying assumptions, and the critical value? Under
Assume that the compressive strength (lb/in.2) of Boston blue clay has a true variance of 15.5 (lb/in.2)2. Can we conclude that the sample of values in Problem 9.61 has a mean that is significantly less than 50 lb/in.2?
Results of testing for the presence of pollutants in a local stream have a mean of 10 mg/L and a standard deviation of 2 mg/L. Six samples of water collected from the stream result in the following measurements: 12.7, 15.1, 9.5, 13.7, 19.6, and 16.4 mg/L. Does the level of pollutants in the stream
Using the first sample of 5 in Table 8.3a, test the null hypothesis that μ = 0. Use a two-sided alternative hypothesis and a 5% level of significance.
Using the sample mean on row 2, column 7 of Table 8.3b, test whether or not it is significantly less than 1.0. Use a one-tailed alternative hypothesis and a 2% level of significance. What is the rejection probability?
Using the sample mean on row 3, column 6 of Table 8.3b, test whether or not it is significantly different from 0. Use a two-sided alternative hypothesis and a 5% level of significance.
From Table 8.3a, find the sample that has a sample mean that deviates from 0 by the largest amount.Test the mean to determine if it is significantly different from 0. Use a two-sided alternative hypothesis and a 5% level of significance. Assuming the samples are from a N(0,1) population, is it
What four factors influence the critical value of a test statistic? Show pictorially how each factor affects the critical value.
Define the region of rejection in terms of (a) values of the test statistic; (b) proportions of the area of the probability density function of the test statistic; (c) the region of acceptance; and (d) the critical value(s) of the test statistic.
A new type of steel is designed for a tensile strength of x. What are the implications of type I and type II errors?
A levee is designed to control a discharge of 800 ft3/s. A model is used to predict discharge rates for various storms. What are the implications of type I and type II errors?
A can of paint is advertised to cover an area of 400 ft2. What are the implications of type I and type II errors?
What factors influence the selection of the level of significance? Discuss each factor.
Provide statements of the null and alternative hypotheses that the correlation coefficient is equal to some prespecified value. Define all notations.
What are the characteristics of a null hypothesis? Alternative hypothesis?
Write and execute a computer program to derive an approximation of the sampling distribution of the variance. Use a random-number generator that generates standard normal deviates with a mean of zero and a variance of one. Generate 1000 samples of size n and compute the variance of each sample.
A random variable X has a standard normal distribution. If many samples of 5 are drawn from the population, what sample variance will be exceeded 5% of the time? How does this compare with the 40 samples in Table 8.3(c)?
A random variable X has a normal population with a mean μ and a variance of 3. (a) If samples of 10 are taken from the population, sketch the distribution of the variance. (b) Approximately what proportion of 10,000 samples, each with a sample size of 10, would have sample variances greater than
Write and execute a computer program to derive an approximation of the sampling distribution of the mean. Use a random-number generator that generates standard normal deviates with a mean of zero and a variance of one. Generate 1000 samples of size n and compute the mean of each sample.Develop a
A normal population is known to have a mean of 10. (a) If the standard deviation of the population is 5, what is the probability that a random sample of size 25 will have a mean of 12 or higher? (b)What is the probability in (a) if n = 100? (c) If the standard deviation of the population is 10,
Twelve measurements of a pollution concentration (X) are made, with a mean of 42 mg/L and a standard deviation of 8 mg/L. Sketch the distributions of X, the mean, and the variance of X.
Nine measurements are made on the roughness of a concrete pipe (e). The mean and standard deviation are 0.018 and 0.002, respectively. Sketch the distributions of e - and the variance of e.
The population standard deviation of the random variable x is equal to 4. Show how the sample mean varies as the sample size varies from 10 to 50 using a population mean of 25.
A geologist who needs an estimate of the porosity of a sandy soil in large field collects 15 specimens, which yields a mean of 31%. Studies have shown that fine mixed sands have a mean of approximately 32% and a standard deviation of 2%. What is the probability that the samples are actually from
The specific conductance (μS/cm) is one measure of the salinity of prairie lakes and wetlands. In a specific wetland, a biogeochemical analysis suggests that the conductance is normally distributed with a mean of 2500 (μS/cm) with a standard deviation of 425 (μS/cm). (a) What is the probability
A supply of resistors that is intended to have a mean of 24 ohms and a standard deviation of no more than 1.5 ohms is manufactured. A sample of 5 resistors has a mean of 25 ohms. What is the probability that a sample of 5 would produce a mean greater than 25?
The efficiency of a certain type of pump has a mean of 72% and a standard deviation of 3.2%. (a)Assuming a normal population, what is the probability of a single pump selected at random having an efficiency of 75% or greater? (b) What is the probability that a sample of 3 pumps drawn at random will
If a random sample of 8 resistors has a mean of 11 ohms and a standard deviation of 2 ohms, sketch the distributions of both the resistance and its mean.
If a random sample of 5 static friction coefficients μ has a mean of 0.005 and a standard deviation of 0.0002, sketch the distributions of both μ and its mean.
A random variable has a population mean of 45 and a standard deviation of 10. What is the probability that a sample mean computed from a sample of 15 drawn from the population will be greater than 47.5?
Use the method of maximum likelihood to derive estimators for the two-parameter exponential function:f(x) = (1/b) exp(−(x − a)/b).
Derive the estimator k for the function f(x) = x/k for 1 ʺ x ʺ 3 using the method of maximum likelihood.
Using the histogram for the daily evaporation data of Example 9.2, evaluate the parameters of a population having a uniform distribution using the method of moments (X = 0.1387, and S = 0.0935).
If the mean is equal 4, use the method of moments to provide estimates for the constants a and b for the density function f(x) = 0.5x/(b − a)2 for a ʺ x ʺb. Graph the resulting f(x).
Use the method of moments to provide an estimator for k for the discrete random variable x:f(x) = 3x2 / k3 for x = 1, 2, 3, 4.
Compute the mean, variance, and standard deviation for a continuous random variable x with the density function fX(x) = 2x/9 over the range from 0 to 3.
A continuous random variable has the following density function: fX(x) = 3x2/26 for the interval (1 ≤x ≤ 3). Find the mean, variance, and standard deviation of x.
Is the following function a legitimate mass function for a discrete random variable x? f (x) = 12 /(13x)for x = 2, 3, 4. If so, compute the mean value.
Show the calculations of the mean and standard deviation for the data of Equation 9.40a. Plot a histogram of the data and show the mean and standard deviation on the graph.
Find the mean, variance, and standard deviation of the eight largest values and separately the eight smallest values of sediment yield given in Table 9.4. Discuss the differences in the statistics.
Compute the mean and standard deviation for the following pH measurements made from water in a lake: 6.4, 5.7, 6.8, 7.3, and 6.3. Discuss whether or not the pH of the water is evenly distributed throughout the lake.
If the true temperature of a body of water is 16.5°C and four of five measurements of the temperature are 16.2, 17.4, 17.1, and 15.9°C, what value of the fifth measurement would be necessary for the measurement method to be unbiased?
Grades for college courses are generally based on scores on hourly tests, quizzes, final exams, homework assignments, and project reports. Discuss bias, precision, and accuracy as they are relevant to final grades in college courses.
Traffic officers use radar guns to catch speeders. Discuss bias, precision, and accuracy as they are relevant to ticketing speeders.
Discuss the concepts of bias, precision, and accuracy in terms of an archer shooting arrows at a target.Sketch targets to illustrate each of the three concepts.
Identify the population from which each of the following samples were drawn: (a) the times of the last five Kentucky Derby winners; (b) the fuel efficiencies (miles/gal) of 100 Toyota Camrys; (c) the grade point averages of 55 graduating seniors; (d) the drag coefficients computed from five wind
With respect to collecting samples of stream water to identify the concentration of dissolved oxygen, discuss the concepts of samples and populations.
Compare the results of Problems 7.34 and 7.35 to the rod in Problems 7.32 and 7.33. Provide a discussion.
For the rod in Problems 7.32 and 7.33, plot a frequency histogram for the deformation of the rod based on 1000 simulation cycles. Suggest a distribution type for the deformation.
For the rod in Problems 7.32 and 7.33, plot two frequency histograms for the deformation of the rod based on 20 and 100 simulation cycles. Suggest a distribution type for the deformation.
For the rod in Problem 7.32, study the effect of increasing the number of simulation cycles on the estimated mean and variance of the deformation. Use the following numbers of simulation cycles:20, 100, 500, 1000, 2000, and 10,000. Provide your results in the form of plots of estimated statistics
Assume the following values of xi are a sample from an exponential distribution: xi = {0.27, 0.62, 1.12, 1.35}. Generate exponential variates that are from the same population using the following uniform variates:ui = { 0.47,0.92,0.23,0.52,0.17}
Using inverse transformation, generate values of exponential variates for the parameter λ = 1.4 with the following uniform variates:ui = { 0.34,0.83,0.02}
The mean (m ) Y of Y = ln(X) and standard deviation (σY) of Y = ln(X) are 2.76 and 0.57, respectively.Generate values of x that have a lognormal distribution using the following uniform variates:ui = { 0.03,0.82,0.37,0.54}
Assume that floods are lognormally distributed with a mean of X of 254 m3/sec and a standard deviation of X of 38 m3/sec. Use the following uniform variates to generate values of x that are lognormally distributed:ui = { 0.39,0.61,0.50,0.13,0.82}
Assume the random variable X has a lognormal distribution and y = log(x) with mean Y of 1.32 and standard deviation S of 0.17. Generate values of x for the following uniform variates:ui = { 0.13,0.71,0.44,0.60,0.27}
What uniform variate ui for U(0,1) was used to generate a normal variate of 7.81 from N(6,32) with mean of 6 and standard deviation of 3?
Generate 20 Poisson variates with parameter λt = 1.5. Use the inverse transformation method and the 20 uniform variates of Problem 7.12. Compare the population and sample probabilities.
A discrete random variable (N) has a geometric distribution with parameter p = 0.5. Using the first ten values of ui in Problem 7.17, generate values of N using the inverse transformation method.Compare these to values generated with Equation 7.23.
Generate 10 binomial variates with p = 0.65 and n = 3 using the first three columns of uniform variates in Table 7.4a. Compare the population and sample probabilities.
The roll of a die is used to simulate the toss of a coin. If the roll produces an even value, a head is assumed. If the roll produces an odd value, a tail is assumed. Indicate the outcomes of coin flips if the random sequence of rolls is {2, 5, 4, 2, 1, 6, 3, 3, 6}.
A random variable x can take on values of 1, 2, 3, 4, and 5. A random variable y can take on values of 2, 4, and 6. If a random sequence of x is {2, 4, 3, 1, 5, 4, 3, 4} and the corresponding sequence of y is {4, 2, 6, 6, 2, 2, 6, 2}, what is the transformation rule?
If a random-number generator is available to generate uniformly distributed, 0 to 1 random numbers,(a) how could it be used to generate rolls of a fair die that has the six discrete values? and (b) how could it be used to generate rolls of a loaded die that has a probability of 0.25 for a 1, 0.35
Perform a parametric analysis of the linear congruential generator given by Equation 7.1 by varying the values ofa, b,c, and I0 to investigate their importance. Based on your results, discuss the importance of these parameters.
Using the linear congruential generator given by Equation 7.1, generate a stream of ten random numbers based on the following model parameters: a = 77, b = 345, c = 26, and I0 = 3.
Using the linear congruential generator given by Equation 7.1, generate a stream of ten random numbers based on the following model parameters: a = 7, b = 5, c = 345, and I0 = 3.
Using the linear congruential generator given by Equation 7.1, generate a stream of ten random numbers based on the following model parameters: a = 7, b = 5, c = 26, and I0 = 3.
=+4. Given the manner in which these data were generated, what is the implication of what you observed in Step 3?What does this suggest about the relationship between number of predictors and sample size?
=+, and se values for each of the models fit in Step 2. Write a few sentences describing what happens to each of these three quantities as additional variables are added to the multiple regression model.
=+2. Fit each of the following regression models:i. y with x1 ii. y with x1 and x2 iii. y with x1 and x2 and x3 iv. y with x1 and x2 and x3 and x4 3. Make a table that gives the R 2, the adjusted R 2
=+. Do the scatterplots look the way you expected based on the way the data were generated?Explain.
=+1. Construct four scatterplots—one of y versus each of x1, x2, x3, and x4
=+find the estimated regression equation and assess the utility of the multiple regression model y 5 a 1 b1x1 1 b2x2 1 e y x1 x2 y x1 x2 61 21.0 57.0 77 24.8 48.0 87 28.3 41.5 93 26.0 56.0 98 27.5 58.0 100 27.1 31.0 104 26.8 36.5 118 29.0 41.0 102 28.3 40.0 74 34.0 25.0 63 30.5 34.0 43 28.3 13.0
=+appeared in the article “Estimation of the Economic Threshold of Infestation for Cotton Aphid” (Mesopotamia Journal of Agriculture [1982]: 71–75). Use the data to
=+14.33 ● This exercise requires the use of a computer package. The cotton aphid poses a threat to cotton crops in Iraq. The accompanying data on y 5 infestation rate (aphids/100 leaves)x1 5 mean temperature (8C)x2 5 mean relative humidity
=+c. Interpret the values of the following quantities:SSResid, R 2, se.
=+b. Using a .05 significance level, perform the model utility test.
=+a. Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms.

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