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statistics for engineers and scientists

Principles Of Statistics For Engineers And Scientists 1st Edition William Navidi - Solutions

4. Incinerators can be a source of hazardous emissions into the atmosphere. Stack gas samples were collected from a sample of 50 incinerators in a major city. Of the 50 samples, only 18 met an environmental standard for the concentration of a hazardous compound. Can it be concluded that fewer than
2. Do patients value interpersonal skills more than technical ability when choosing a primary care physician? The article “Patients’ Preferences for Technical Versus Interpersonal Quality When Selecting a Primary Care Physician” (C. Fung, M. Elliot, et al., Health Services Research,
1. Gravel pieces are classified as small, medium, or large. A vendor claims that at least 10% of the gravel pieces from her plant are large. In a random sample of 1600 pieces, 130 pieces were classified as large. Is this enough evidence to reject the claim?
20. Refer to Exercise 18. It is discovered that the standard deviation of the sample used to compute the confidence interval is 5 N. Is it possible to determine whether P < 0.01? Explain.
15. Refer to Exercise 14. For which null hypothesis will P = 0.05? i. H0 :µ = 1.2 ii. H0 :µ ≤ 1.2 iii. H0 :µ ≥ 1.2
9. In each of the following situations, state the most appropriate null hypothesis regarding the population mean µ.a. A new type of epoxy will be used to bond wood pieces if it can be shown to have a mean shear stress greater than 10 MPa.b. A quality control inspector will recalibrate a flowmeter
7. The following MINITAB output presents the results of a hypothesis test for a population mean µ. One-Sample Z: X Test of mu = 37 vs not = 37 The assumed standard deviation = 3.2614 Variable N Mean StDev SE Mean 95% CI Z P X 87 36.5280 3.2614 0.3497 (35.8247, 37.2133) −1.35 0.177a. Can H0 be
6. George performed a hypothesis test. Luis checked George’s work by redoing the calculations. Both George and Luis agree that the result was statistically significant the 5% level, but they got different P-values. George got a P-value of 0.20, and Luis got a P-value of 0.02.a. Is is possible
5. H0 is rejected at the 5% level. True or false:a. The result is statistically significant at the 10% level.b. The result is statistically significant at the 5% level.c. The result is statistically significant at the 1% level.
1. For which P-value is the null hypothesis more plausible: P = 0.10 or P = 0.01?
15. The following MINITAB output presents the results of a hypothesis test for a population mean µ. Some of the numbers are missing. Fill them in. One-Sample Z: X Test of mu = 3.5 vs > 3.5 The assumed standard deviation = 2.00819 95% Lower Variable N Mean StDev SE Mean Bound Z P X 87 4.07114
14. The following MINITAB output presents the results of a hypothesis test for a population mean µ. One-Sample Z: X Test of mu = 73.5 vs not = 73.5 The assumed standard deviation = 2.3634 Variable N Mean StDev SE Mean 95% CI Z P X 145 73.2461 2.3634 0.1963 (72.8614, 73.6308) −1.29 0.196a. Is
13. Fill volumes, in oz, of a large number of beverage cans were measured, with X = 11.98 and σX = 0.02. Use this information to find the P-value for testing H0 :µ = 12.0 versus H1 :µ = 12.0.
10. A new concrete mix is being designed to provide adequate compressive strength for concrete blocks. The specification for a particular application calls for the blocks to have a mean compressive strength µ greater than 1350 kPa. A sample of 100 blocks is produced and tested. Their mean
9. In an experiment to measure the lifetimes of parts manufactured from a certain aluminum alloy, 67 parts were loaded cyclically until failure. The mean number of kilocycles to failure was 763, and the standard deviation was 120. Let µ represent the mean number of kilocycles to failure for
6. A certain type of stainless steel powder is supposed to have a mean particle diameter of µ = 15 µm. A random sample of 87 particles had a mean diameter of 15.2 µm, with a standard deviation of 1.8 µm. A test is made of H0 :µ = 15 versus H1 :µ = 15.a. Find the P-value.b. Do you believe it
5. There is concern that increased industrialization may be increasing the mineral content of river water. Ten years ago, the silicon content of the water in a certain river was 5 mg/L. Eighty-five water samples taken recently from the river have mean silicon content 5.4 mg/L and standard deviation
4. In a process that manufactures tungsten-coated silicon wafers, the target resistance for a wafer is 85 m2. In a simple random sample of 50 wafers, the sample mean resistance was 84.8 m2, and the standard deviation was 0.5 m2. Let represent the mean resistance of the wafers manufactured by this
1. Recently many companies have been experimenting with telecommuting, allowing employees to work at home on their computers. Among other things, telecommuting is supposed to reduce the number of sick days taken. Suppose that at one firm, it is known that over the past few years employees have
21. Let X represent the number of events that are observed to occur in n units of time or space, and assume X ∼ Poisson(nλ), where λ is the mean number of events that occur in one unit of time or space. Assume X is large, so that X ∼ N(nλ, nλ). Follow steps (a) through (d) to derive a
19. The temperature of a certain solution is estimated by taking a large number of independent measurements and averaging them. The estimate is 37◦ C, and the standard error is 0.1◦ C.a. Find a 95% confidence interval for the temperature.b. What is the confidence level of the interval 37 ±
14. A 95% confidence interval for a population mean is computed from a sample of size 400. Another 95% confidence interval will be computed from a sample of size 100 drawn from the same population. Choose the best answer to fill in the blank: The interval from the sample of size 400 will be
10. Refer to Exercise 9. A device will be manufactured in which two of the components in Exercise 9 will be connected in series. The components function independently, and the device will function only if both components function. Let q be the probability that a device functions. Find a 95%
7. A 90% confidence interval for a population mean based on 144 observations is computed to be (2.7, 3.4). How many observations must be made so that a 90% confidence interval will specify the mean to within ±0.2?
6, and the standard deviation was 3.2.a. Find a 90% confidence interval for the mean porosity of specimens of this type of concrete.b. Find a 95% confidence interval for the mean porosity of specimens of this type of concrete.c. What is the confidence level of the interval (21.0, 22.2)?d. How
5. A survey is to be conducted in which a random sample of residents in a certain city will be asked whether they favor or oppose a modernization project for the civic center. How many residents should be polled to be sure that a 98% confidence interval for the proportion who are in favor specifies
2. In a sample of 150 boards of a certain grade, the average modulus of elasticity was 1.57 psi, and the standard deviation was 0.23 psi.a. Find a 95% confidence interval for the mean modulus of elasticity.b. Find a 99.5% confidence interval for the mean modulus of elasticity.c. What is the
1. Concentrations of atmospheric pollutants such as carbon monoxide (CO) can be measured with a spectrophotometer. In a calibration test, 50 measurements were taken of a laboratory gas sample that is known to have a CO concentration of 70 parts per million (ppm). A measurement is considered to be
5. Five measurements are taken of the octane rating for a particular type of gasoline. The results (in %) are 87.0, 86.0, 86.5, 88.0, 85.3. These data also appear in Exercise 4 in Section 5.4.a. Find a 95% prediction interval for a single measurement.b. Find a tolerance interval for the pH that
3. The article “Ozone for Removal of Acute Toxicity from Logyard Run-off” (M. Zenaitis and S. Duff, Ozone Science and Engineering, 2002:83–90) presents chemical analyses of runoff water from sawmills in British Columbia. Included were measurements of pH for six water specimens: 5.9, 5.0,
2. In a sample of 20 bolts, the average breaking torque was 89.7 J with a standard deviation of 8.2 J. Assume that the breaking torques are normally distributed.a. Find a 99% prediction interval for the breaking torque of a single bolt.b. Find a tolerance interval for the breaking torque that
1. A sample of 25 resistors, each labeled 100 , had an average resistance of 101.4 with a standard deviation of 2.3 . Assume the resistances are normally distributed.a. Find a 95% prediction interval for the resistance of a single resistor.b. Find a tolerance interval for the resistance that
17. The article “Filtration Rates of the Zebra Mussel (Dreissena polymorpha) on Natural Seston from Saginaw Bay, Lake Huron” (D. Fanslow, T. Nalepa, and G. Lang, Journal of Great Lakes Research, 1995:489–500) reports measurements of the rates (in mL/mg/h) at which mussels filter seston
16. The concentration of carbon monoxide (CO) in a gas sample is measured by a spectrophotometer and found to be 85 ppm. Through long experience with this instrument, it is believed that its measurements are unbiased and normally distributed, with an standard deviation of 8 ppm. Find a 95%
15. The following MINITAB output presents a confidence interval for a population mean, but some of the numbers got smudged and are now illegible. Fill in the missing numbers. One-Sample T: X Variable N Mean StDev SE Mean 99% CI X 20 2.39374 (a) 0.52640 ( (b), (c) )
13. In an experiment to measure the rate of absorption of pesticides through skin, 500 µg of uniconazole was applied to the skin of four rats. After 10 hours, the amounts absorbed (in µg) were 0.5, 2.0, 1.4, and 1.1. Find a 90% confidence interval for the mean amount absorbed.
12. Surfactants are chemical agents, such as detergents, that lower the surface tension of a liquid. Surfactants play an important role in the cleaning of contaminated soils. In an experiment to determine the effectiveness of a certain method for removing toluene from sand, the sand was washed with
11. The article “An Automatic Visual System for Marble Tile Classification” (L. Carrino, W. Polini, and S. Turchetta, Journal of Engineering Manufacture, 2002:1095–1108) describes a measure for the shade of marble tile in which the amount of light reflected by the tile is measured on a scale
10. The following are summary statistics for a data set. Would it be appropriate to use the Student’s t distribution to construct a confidence interval from these data? Explain. N Mean Median StDev 10 8.905 6.105 9.690 Minimum Maximum Q1 Q3 0.512 39.920 1.967 8.103
6. A chemist made eight independent measurements of the melting point of tungsten. She obtained a sample mean of 3410.14◦ C and a sample standard deviation of 1.018◦ C.a. Use the Student’s t distribution to find a 95% confidence interval for the melting point of tungsten.b. Use the
5. A model of heat transfer from a cylinder immersed in a liquid predicts that the heat transfer coefficient for the cylinder will become constant at very low flow rates of the fluid. A sample of 10 measurements is taken. The results, in W/m2 K, are 13.7 12.0 13.1 14.1 13.1 14.1 14.4 12.2 11.9 11.8
4. Five measurements are taken of the octane rating for a particular type of gasoline. The results (in %) are 87.0, 86.0, 86.5, 88.0, 85.3. Find a 99% confidence interval for the mean octane rating for this type of gasoline.
3. Find the level of a two-sided confidence interval that is based on the given value of tn−1,α/2 and the given sample size.a. t = 2.179, sample size 13.b. t = 3.365, sample size 6.c. t = 1.729, sample size 20.d. t = 3.707, sample size 7.e. t = 3.707, sample size 27.
2. Find the value of tn−1,α needed to construct an upper or lower confidence bound in each of the situations in Exercise 1.
1. Find the value of tn−1,α/2 needed to construct a twosided confidence interval of the given level with the given sample size:a. Level 90%, sample size 9.b. Level 95%, sample size 5.c. Level 99%, sample size 29.d. Level 95%, sample size 2.
10. A voltmeter is used to record 100 independent measurements of a known standard voltage. Of the 100 measurements, 85 are within 0.01 V of the true voltage.a. Find a 95% confidence interval for the probability that a measurement is within 0.01 V of the true voltage.b. Find a 98% confidence
8. Refer to Exercise 3. Find a 99% lower confidence bound for the proportion of gasoline stations with at least one leaking underground tank.
6. Refer to Exercise 1. Find a 95% lower confidence bound for the proportion of automobiles whose emissions exceed the standard.
4. In a random sample of 150 households with an Internet connection, 32 said that they had changed their Internet service provider within the past six months.a. Find a 95% confidence interval for the proportion of customers who changed their Internet service provider within the past six months.b.
3. Leakage from underground fuel tanks has been a source of water pollution. In a random sample of 87 gasoline stations, 13 were found to have at least one leaking underground tank.a. Find a 95% confidence interval for the proportion of gasoline stations with at least one leaking underground
2. During a recent drought, a water utility in a certain town sampled 100 residential water bills and found that 73 of the residences had reduced their water consumption over that of the previous year.a. Find a 95% confidence interval for the proportion of residences that reduced their water
1. In a simple random sample of 70 automobiles registered in a certain state, 28 of them were found to have emission levels that exceed a state standard.a. What proportion of the automobiles in the sample had emission levels that exceed the standard?b. Find a 95% confidence interval for the
18. Refer to Exercise 11.a. Find a 95% upper confidence bound for the mean breaking strength.b. The supplier claims that the mean breaking strength is greater than 28 lb. With what level of confidence can this statement be made?
16. Refer to Exercise 9.a. Find a 90% upper confidence bound for the mean weight.b. Someone says that the mean weight is less than 1.585 g. With what level of confidence can this statement be made?
15. Refer to Exercise 8.a. Find a 99% upper confidence bound for the concentration.b. The claim is made that the concentration is less than 1.75 ppb. With what level of confidence can this statement be made?
14. Refer to Exercise 7.a. Find a 95% lower confidence bound for the mean compressive strength of this type of box.b. An engineer claims that the mean compressive strength is greater than 6220 N. With what level of confidence can this statement be made?
13. Refer to Exercise 6.a. Find a 98% lower confidence bound for the mean resistance.b. An engineer says that the mean resistance is greater than 17 m. With what level of confidence can this statement be made?
12. Refer to Exercise 5.a. Find a 95% lower confidence bound for the mean capacity of this type of battery.b. An engineer claims that the mean capacity is greater than 175 hours. With what level of confidence can this statement be made?
10. One step in the manufacture of a certain metal clamp involves the drilling of four holes. In a sample of 150 clamps, the average time needed to complete this step was 72 seconds and the standard deviation was 10 seconds.a. Find a 95% confidence interval for the mean time needed to complete the
8. Polychlorinated biphenyls (PCBs) are a group of synthetic oil-like chemicals that were at one time widely used as insulation in electrical equipment and were discharged into rivers. They were discovered to be a health hazard and were banned in the 1970s. Since then, much effort has gone into
7. In a sample of 100 boxes of a certain type, the average compressive strength was 6230 N, and the standard deviation was 221 N.a. Find a 95% confidence interval for the mean compressive strength of boxes of this type.b. Find a 99% confidence interval for the mean compressive strength of boxes
6. Resistance measurements were made on a sample of 81 wires of a certain type. The sample mean resistance was 17.3 m, and the standard deviation was 1.2 m.a. Find a 95% confidence interval for the mean resistance of this type of wire.b. Find a 98% confidence interval for the mean resistance
4. Interpolation methods are used to estimate heights above sea level for locations where direct measurements are unavailable. In the article “Transformation of Ellipsoid Heights to Local Leveling Heights” (M. Yanalak and O. Baykal, Journal of Surveying Engineering, 2001:90–103), a
2. Find the levels of the confidence intervals that have the following values of zα/2.a. zα/2 = 1.96b. zα/2 = 2.81c. zα/2 = 2.17d. zα/2 = 2.33
1. Find the value of zα/2 to use in expression (5.4) to construct a confidence interval with levela. 90%b. 83%c. 99.5%d. 75%
5. Refer to Exercise 3. For what values of µ does ˆµ3 have smaller mean squared error than ˆµ2?
4 . Find the bias, variance, and mean squared error of ˆµ3. 4. Refer to Exercise 3. For what values of µ does ˆµ3 have smaller mean squared error than ˆµ1?
3 . Find the bias, variance, and mean squared error of ˆµ2.c. Let ˆµ3 = X1 + X2
2 . Find the bias, variance, and mean squared error of ˆµ1.b. Let ˆµ2 = X1 + 2X2
1. Choose the best answer to fill in the blank. If an estimator is unbiased, then i. the estimator is equal to the true value. ii. the estimator is usually close to the true value. iii. the mean of the estimator is equal to the true value. iv. the mean of the estimator is usually close to the true
12. A process for manufacturing plate glass leaves an average of three small bubbles per 10 m2 of glass. The number of bubbles on a piece of plate glass follows a Poisson distribution.a. What is the probability that a piece of glass 3 × 5 m will contain more than two bubbles?b. What is the
11. In a certain process, the probability of producing an oversize assembly is 0.05.a. In a sample of 300 randomly chosen assemblies, what is the probability that fewer than 20 of them are oversize?b. In a sample of 10 randomly chosen assemblies, what is the probability that one or more of them is
8. You have a large box of resistors whose resistances are normally distributed with mean 10 and standard deviation 1 .a. What proportion of the resistors have resistances between 9.3 and 10.7 ?b. If you sample 100 resistors, what is the probability that 50 or more of them will have
6. In the article “Occurrence and Distribution of Ammonium in Iowa Groundwater” (K. Schilling, Water Environment Research, 2002:177–186), ammonium concentrations (in mg/L) were measured at a large number of wells in Iowa. The mean concentration was 0.71, the median was 0.22, and the
9. A sample of 225 wires is drawn from the population of wires described in Example 4.30. Find the probability that fewer than 110 of these wires have no flaws.
5. In a process that manufactures bearings, 90% of the bearings meet a thickness specification. In a sample of 500 bearings, what is the probability that more than 440 meet the specification?
3. In a galvanized coating process for pipes, the mean coating weight is 125 lb, with a standard deviation of 10 lb. A random sample of 50 pipes is drawn.a. What is the probability that the sample mean coating weight is greater than 128 lb?b. Find the 90th percentile of the sample mean coating
6. Construct a normal probability plot for the logs of the PM data in Table 1.2. Do the logs of the PM data appear to come from a normal population?
5. Construct a normal probability plot for the PM data in Table 1.2 (page 17). Do the PM data appear to come from a normal population?
4. Below are the durations (in minutes) of 40 time intervals between eruptions of the geyser Old Faithful in Yellowstone National Park. 91 51 79 53 82 51 76 82 84 53 86 51 85 45 88 51 80 49 82 75 73 67 68 86 72 75 75 66 84 70 79 60 86 71 67 81 76 83 76 55 Construct a normal probability plot for
3. Below are the durations (in minutes) of 40 eruptions of the geyser Old Faithful in Yellowstone National Park.4.1 1.8 3.2 1.9 4.6 2.0 4.5 3.9 4.3 2.3 3.8 1.9 4.6 1.8 4.7 1.8 4.6 1.9 3.5 4.0 3.7 3.7 4.3 3.6 3.8 3.8 3.8 2.5 4.5 4.1 3.7 3.8 3.4 4.0 2.3 4.4 4.1 4.3 3.3 2.0 Construct a normal
2. Construct a normal probability plot for the soapbars data in Exercise 1 in Section 1.3 (page 30). Do these data appear to come from an approximately normal distribution?
1. Each of three samples has been plotted on a normal probability plot. For each, say whether the sample appears to have come from an approximately normal population. 0.999 0.99 0.95 0.999 0.99 0.95 0.9 0.75 0.5 0.25 0.1 0.05 0.01 0.001 0.9 0.75 0.5 0.25 0.1 0.05 (a) 0.01 0.001 0.999 0.99 0.95 0.9
6. If T is a continuous random variable that is always positive (such as a waiting time), with probability density function f (t) and cumulative distribution function F(t), then the hazard function is defined to be the function h(t) = f (t) 1 − F(t) The hazard function is the rate of failure per
5. Let T ∼ Weibull(0.5, 2).a. Find µT .b. Find σ2 T .c. Find P(T ≤ 2).d. Find P(T > 3).e. Find P(1 < T ≤ 2).
4. Let T ∼ (r, λ). If µT = 8 and σ2 T = 16, find r and λ.
3. Let T ∼ (6, 2).a. Find µT .b. Find σT .
2. Resistors are labeled 100 . In fact, the actual resistances are uniformly distributed on the interval (95, 103).a. Find the mean resistance.b. Find the standard deviation of the resistances.
1. A person arrives at a certain bus stop each morning. The waiting time, in minutes, for a bus to arrive is uniformly distributed on the interval (0, 10).a. Find the mean waiting time.b. Find the standard deviation of the waiting times.
3. A catalyst researcher states that the diameters, in microns, of the pores in a new product she has made follow the exponential distribution with parameter λ = 0.25.a. What is the mean pore diameter?b. What is the standard deviation of the pore diameters?c. What proportion of the pores are
1. Let T ∼ Exp(0.5). Finda. µTb. σ2 Tc. P(T > 5)d. The median of T
2. The article “Assessment of Dermopharmacokinetic Approach in the Bioequivalence Determination of Topical Tretinoin Gel Products” (L. Pershing, J. Nelson, et al., Journal of The American Academy of Dermatology, 2003:740–751) reports that the amount of a certain antifungal ointment that is
9. The strength of an aluminum alloy is normally distributed with mean 10 gigapascals (GPa) and standard deviation 1.4 GPa.a. What is the probability that a specimen of this alloy will have a strength greater than 12 GPa?b. Find the first quartile of the strengths of this alloy.c. Find the 95th
8. At a certain university, math SAT scores for the entering freshman class averaged 650 and had a standard deviation of 100. The maximum possible score is 800. Is it possible that the scores of these freshmen are normally distributed? Explain.
7. The lifetime of a light bulb in a certain application is normally distributed with mean µ = 1400 hours and standard deviation σ = 200 hours.a. What is the probability that a light bulb will last more than 1800 hours?b. Find the 10th percentile of the lifetimes.c. A particular battery lasts
5. Scores on a standardized test are approximately normally distributed with a mean of 460 and a standard deviation of 80.a. What proportion of the scores are above 550?b. What is the 35th percentile of the scores?c. If someone’s score is 600, what percentile is she on?d. What proportion of the
4. If X ∼ N(3, 4), computea. P(X ≥ 3)b. P(1 ≤ X < 8)c. P(−1.5 ≤ X < 1)d. P(−2 ≤ X − 3 < 4)
3. Let Z ∼ N(0, 1). Find a constant c for whicha. P(Z ≤c) = 0.8413b. P(0 ≤ Z ≤c) = 0.3051c. P(−c ≤ Z ≤c) = 0.8664d. P(c ≤ Z ≤ 0) = 0.4554e. P(|Z| ≥c) = 0.1470
2. Find the area under the normal curvea. To the right of z = 0.73.b. Between z = 2.12 and z = 2.57.c. Outside z = 0.96 to z = 1.62.d. Between z = −0.13 and z = 0.70.
1. Find the area under the normal curvea. To the right of z = −0.75.b. Between z = 0.40 and z = 1.15.c. Between z = −0.60 and z = 0.70.d. Outside z = 0.75 to z = 1.30.

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