Refer to the Environmental Science & Technology (Jan. 2005) study of methyl tert-butyl ether (MTBE) contamination in New Hampshire wells, Exercise 7.66. How many public and how many private wells must be sampled in order to estimate the difference between the proportions of wells with a detectable level of MTBE to within .06 with 95% confidence?
Data from Exercise 7.66
Refer to the Environmental Science & Technology (Jan. 2005) study of methyl tert-butyl ether (MTBE) contamination in New Hampshire wells, Exercise 2.12. Data collected for a sample of 223 wells are saved in the MTBE file. Recall that each well was classified according to well class (public or private) and detectable level of MTBE (below limit or detect). The accompanying SPSS printout gives the number of wells in the sample with a detectable level of MTBE for both the 120 public wells and the 103 private wells.
Data from Exercise 2.12
In New Hampshire, about half the counties mandate the use of reformulated gasoline. This has led to an increase in the contamination of groundwater with methyl tert-butyl ether (MTBE). Environmental Science & Technology (Jan. 2005) reported on the factors related to MTBE contamination in public and private New Hampshire wells. Data were collected for a sample of 223 wells. These data are saved in the MTBE file. Three of the variables are qualitative in nature: well class (public or private), aquifer (bedrock or unconsolidated), and detectable level of MTBE (below limit or detect). (Note: A detectable level of MTBE occurs if the MTBE value exceeds .2 micrograms per liter.) The data for 10 selected wells are shown in the accompanying table. Use graphical methods to describe each of the three qualitative variables for all 223 wells.
Chemical engineers at the University of Murcia (Spain) conducted a series of experiments to determine the most effective membrane to use in a passive sampler (Environmental Science & Technology, Vol. 27, 1993). The effectiveness of a passive sampler was measured by the sampling rate, recorded in cubic centimeters per minute. In one experiment, six passive samplers were positioned with their faces parallel to the air flow and with an air velocity of 90 centimeters per second. After 6 hours, the sampling rate of each was determined. Based on the results, a 95% confidence interval for the mean sampling rate was calculated to be (49.66, 51.48).
a. What is the confidence coefficient for the interval?
b. Give a theoretical interpretation of the confidence coefficient
c. Give a practical interpretation of the confidence interval. d. What assumptions, if any, are required for the interval to yield valid inferences?
Refer to the Environmental Science & Technology (Jan. 2005) study of contaminated well sites located near a New Jersey gasoline service station, Exercise 7.29. The data (parts per billion) on the level of methyl tert-butyl ether (MTBE) at 12 sampled sites are reproduced in the table. Use the bootstrap procedure to estimate the mean MTBE level for all well sites located near the New Jersey gasoline service station with a 99% confidence interval. Compare the bootstrap interval with the interval you obtained in Exercise 7.29, part b.
Data from Exercise 7.29
Methyl t-butyl ether (MTBE) is an organic water contaminant that often results from gasoline spills. The level of MTBE (in parts per billion) was measured for a sample of 12 well sites located near a gasoline service station in New Jersey. (Environmental Science & Technology, Jan. 2005.)
In fields that require high precision surface height maps, white light interferometry (WLI) has become the standard inspection tool. Because WLI generates two-dimensional height profiles, and standard mechanical devices used with WLI generate only one-dimensional profiles, engineers must estimate the mean height of pixels generated in WLI surface maps. In Optical Engineering (Jan. 2005), German researchers applied Bayesian estimation to solve the problem. A simplified version of the research is stated as follows: Let Y represent the height for a pixel generated by WLI. Assume that Y takes on the value 1 with probability p and the value 0 with probability (1 – p). Also, assume that p has a beta distribution with parameters α = 1 and β = 2. Now let y1, y2, y3, . . . , yn represent the heights for a sample of n pixels. Using a squared error loss function, find the Bayesian estimate of p if y̅ = .80
Refer to the Journal of Food Engineering (Sep. 2013) study of the characteristics of fried sweet potato chips, Exercise 7.78. Recall that a sample of 6 sweet potato slices fried at 130º using a vacuum fryer yielded the following statistics on internal oil content (measured in gigagrams): y̅1 = .178 g/g and s1 = .011 g/g. A second sample of 6 sweet potato slices was obtained, only these were subjected to a two-stage frying process (again, at 130º) in an attempt to improve texture and appearance. Summary statistics on internal oil content for this second sample follows: y̅2 = .140 g/g and s2 = .002 g/g. The researchers want to compare the mean internal oil contents of sweet potato chips fried with the two methods; however, they recognize that the sample sizes are small.
a. What assumption about the data is required in order for the comparison of means to be valid?
b. Construct a 95% confidence interval for the ratio of the two population variances of interest.
c. Based on the interval, part b, is there a violation of the assumption, part a? Explain.
Data from Exercise 7.78
Refer to the IEICE Transactions on Information & Systems (Jan. 2005) experiment to monitor the impedance to leg movements, Exercise 7.34. Engineers attached electrodes to the ankles and knees of volunteers and measured the signal-to-noise ratio (SNR) of impedance changes. Recall that for a particular ankle–knee electrode pair, a sample of 10 volunteers had SNR values with a mean of 19.5 and a standard deviation of 4.7. Form a 95% confidence interval for the true standard deviation of the SNR impedance changes. Interpret the result.
Refer to the American Journal of Science (Jan. 2005) study of the chemical makeup of buried glacial drifts (or tills) in Wisconsin, Exercise 7.64. The data on the ratios of aluminum (Al) to beryllium (Be) in sediment for a sample of 26 buried till specimens is reproduced in the table.
a. Use the bootstrap procedure to estimate the true proportion of till specimens with an Al/Be ratio that exceeds 4.5 using a 95% confidence interval.
b. Compare the bootstrap interval with the interval you obtained in Exercise 7.56, part b. Why might the bootstrap interval be more appropriate?
Data from Exercise 2.22
Tills are glacial drifts consisting of a mixture of clay, sand, gravel, and boulders. Engineers from the University of Washington’s Department of Earth and Space Sciences studied the chemical makeup of buried tills in order to estimate the age of the glacial drifts in Wisconsin. (American Journal of Science, Jan. 2005.) The ratio of the elements aluminum (Al) and beryllium (Be) in sediment is related to the duration of burial. The Al/Be ratios for a sample of 26 buried till specimens are given in the table. With the aid of a graph, estimate the proportion of till specimens with an Al/Be ratio that exceeds 4.5.
Refer to the JOM (Jan. 2003) comparison of a new high-strength RAA aluminum alloy to the current strongest aluminum alloy, Exercise 7.43. Suppose the researchers want to estimate the difference between the mean yield strengths of the two alloys to within 15 MPa using a 95% confidence interval. How many alloy specimens of each type must be tested in order to obtain the desired estimate?
Data from Exercise 7.43
Mechanical engineers have developed a new high-strength aluminum alloy for use in antisubmarine aircraft, tankers, and long-range bombers. (JOM, Jan. 2003.) The new alloy is obtained by applying a retrogression and reaging (RAA) heat treatment to the current strongest aluminum alloy. A series of strength tests were conducted to compare the new RAA alloy to the current strongest alloy. Three specimens of each type of aluminum alloy were produced and the yield strength (measured in mega-pascals, MPa) of each specimen determined. The results are summarized in the table.
Refer to the Journal of Chemical Ecology (March 2013) study of the effectiveness of pheromones to attract two different strains of fall armyworms, Exercise 7.68. Recall that both corn-strain and rice-strain male armyworms were released into a corn field containing the pheromone and the percentage of males trapped by the pheromone for the two different strains was compared. If the researchers want to estimate the difference in percentages to within 5% with a 90% confidence interval, how many armyworms of each strain need to be released into the field? Assume an equal number of corn-strain and rice-strain males will be released.
Data from Exercise 7.68
A study was conducted to determine the effectiveness of pheromones produced by two different strains of fall armyworms — the corn-strain and the rice-strain (Journal of Chemical Ecology, March 2013). Both corn-strain and rice-strain male armyworms were released into a field containing a synthetic pheromone made from a corn-strain blend. A count of the number of males trapped by the pheromone was then determined. The experiment was conducted once in a corn field, then again in a grass field. The results are provided in the table below:
Refer to the Environmental Geology (Vol. 58, 2009) simulation study of how far a block from a collapsing rock wall will bounce down a soil slope, Exercise 2.29. Rebound lengths (in meters) were estimated for 13 rock bounces. The data are repeated in the table below. A MINITAB analysis of the data is shown in the printout on p. 339.
a. Locate a 95% confidence interval for σ2 on the printout. Interpret the result.
b. Locate a 95% confidence interval for σ on the printout. Interpret the result.
c. What conditions are required for the intervals, parts a and b, to be valid?
Data from Exercise 2.29
In Environmental Geology (Vol. 58, 2009) computer simulation was employed to estimate how far a block from a collapsing rock wall will bounce—called rebound length—down a soil slope. Based on the depth, location, and angle of block-soil impact marks left on the slope from an actual rock fall, the following 13 rebound lengths (meters) were estimated. Compute the mean and median of the rebound lengths and interpret these values.
Refer to Exercise 7.35. Suppose that we want to estimate the average decay rate of fine particles produced from oven cooking or toasting to within .04 with 95% confidence. How large a sample should be selected?
Data from Exercise 7.35
A group of Harvard University School of Public Health researchers studied the impact of cooking on the size of indoor air particles. (Environmental Science & Technology, September 1, 2000.) The decay rate (measured as mm/hour) for fine particles produced from oven cooking or toasting was recorded on six randomly selected days. These six measurements are: