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Probability And Statistics For Engineers And Scientists 3rd Edition Anthony Hayter - Solutions

Consider a sample X1,..., Xn of normally distributed random variables with variance σ2 = 5. Suppose that n = 31. (a) What is the value of c for which P(S2 ≤ c) = 0.90? (b) What is the value of c for which P(S2 ≤ c) = 0.95?
Repeat Problem 7.3.4 with n = 21 and σ2 = 32. Problem 7.3.4 Consider a sample X1,..., Xn of normally distributed random variables with variance σ2 = 5. Suppose that n = 31. (a) What is the value of c for which P(S2 ≤ c) = 0.90? (b) What is the value of c for which P(S2 ≤ c) = 0.95?
Consider a sample X1,..., Xn of normally distributed random variables with mean Î¼. Suppose that n = 16.(a) What is the value of c for which(b) What is the value of c for which
Consider a sample X1,..., Xn of normally distributed random variables with mean Î¼. Suppose that n = 21.(a) What is the value of c for which(b) What is the value off for which
In a consumer survey, 234 people out of a representative sample of 450 people say that they prefer product A to product B. Let p be the proportion of all consumers who prefer product A to product B. Construct a point estimate of p. What is the standard error of your point estimate?
The breaking strengths of 35 pieces of cotton thread are measured. The sample mean is = 974.3 and the sample variance is s2 = 452.1. Construct a point estimate of the average breaking strength of this type of cotton thread. What is the standard error of your point estimate?
Suppose that 23 observations are collected from a Poisson distribution, and the sample average is = 5.63. Construct a point estimate of the parameter of the Poisson distribution and calculate its standard error.
Suppose that a set of observations is collected from a beta distribution, with an average of = 0.782 and a variance of s2 = 0.0083. Obtain point estimates of the parameters of the beta distribution.
Consider a set of independent data observations x1,...,xn that have an exponential distribution with an unknown parameter λ. Show that the method of moments and maximum likelihood estimation both produce the point estimate = 1 /
Consider a set of independent data observations x1,...,xn that have a gamma distribution with k = 5 and an unknown parameter λ. Show that the method of moments and maximum likelihood estimation both produce the point estimate= 5 /
Suppose that X1 and X2 are independent random variables withE(X1) = E(X2) = Î¼andVar(X1) = Var(X2) = 1Show that the point estimate Xx + X2has a smaller mean square error than the point estimate when |Î¼ - 10| ‰¤ ˆš6/2 Why would you expect 1 to have a
Consider the data set of bamboo shoot heights given in DS 6.6.5. Construct a point estimate of the average height. What is the standard error of your point estimate?
Consider the usual point estimates s21 and s22 of the variance σ2 of a normal distribution based on sample sizes n1 and n2, respectively. What is the relative efficiency of the point estimate s21 to the point estimate s22?
Suppose that among 24,839 customers of a certain company, exactly 11.842 feel "very satisfied" with the service they received. In order to estimate the satisfaction levels of the customers, a manager contacts a random sample of 80 of these customers and finds out how main of them were "very
The viscosities of chemical infusions obtained from a specific production technique are normally distributed with a standard deviation σ = 3.9. If a chemist is able to measure the viscosities of 15 independent samples of the infusions, what is the probability that the resulting point estimate of
Recall the data set of soil compressibility measurements given in DS 6.6.6. Construct a point estimate of the average soil compressibility, and find its standard error. What is a point estimate of the upper quartile of the distribution of soil compressibilities?
Suppose that an engineer wishes to estimate the proportion of defective products from a production line. A random sample of 220 products are tested, of which 39 are found to be defective. What is the standard error of the engineer's estimate of the proportion of defective products?
The probability that a medical treatment is effective is 0.68, unknown to a researcher. In an experiment to investigate the effectiveness of the treatment, the researcher applies the treatment in 140 cases and measures whether the treatment is effective or not. What is the probability that the
The biomass of 12 samples was measured, and the following values were obtained:(a) What is the estimate of the mean biomass?(b) What is the standard error of the estimate of the mean biomass?(c) What is the sample median?
Suppose that X ~ B(12, p) and consider the point estimate = X/14 (a) What is the bias of this point estimate? (b) What is the variance of this point estimate? (c) Show that this point estimate has a mean square error of 3p - 2p2 / 49 (d) Show that this mean square error is smaller than the mean
Are the following statements true or false? (a) Statistical inference uses the results of an experiment to make inferences on some properties of an unknown underlying probability distribution. (b) The margin of error in a political poll is based on the standard error of the estimate obtained. (c)
Components have lengths that are independently distributed as a normal distribution with μ = 723 and σ = 3. If an experimenter measures the lengths of a random sample of 11 components, what is the probability that the experimenter's estimate of μ will be between 722 and 724?
An experimenter wishes to estimate the mean weight of some components where the weights have a normal distribution with a standard deviation of 40.0.(a) If the experimenter has a sample size of 10, what is the probability that the estimate is within 20.0 of the correct value?(b) What is the
In a political poll, responses were obtained from a sample of 1962 people about which candidate they preferred.There were 852 people who reported that they preferred candidate A. What is the estimate of the proportion of the overall electorate who prefer candidate A? What is the standard error of
Let X1,..., Xn be a set of independent random variables with a U(0, Î¸) distribution, and letT = max{X1,..., Xn}(a) Explain why the cumulative distribution function of T isF(t) = (t/Î¸)nFor 0 ‰¤ t ‰¤ Î¸.(b) Show that the probability
As in Problem 7.6.3, let X1,..., Xn be a set of independent random variables with a U (0, θ) distribution, and letT = max{X1,..., Xn)Explain why the likelihood function L(x1,..., xn, θ) is equal to1/θnif θ ≥ t = max{x1,..., xn), and is equal to 0 otherwise. Sketch the likelihood function
Consider a set of independent data observations x1,..., xn that have a geometric distribution with an unknown parameter p. Show that the method of moments and maximum likelihood estimation both produce the point estimate = 1 /
Consider the data set of bird species given in DS 6.6.1. Construct a point estimate of the probability that a bird has black markings. What is the standard error of your point estimate?
Consider the data set of monthly accidents given in DS 6.6.2. Construct a point estimate of the average number of accidents per month. What is the standard error of your point estimate?
Consider the data set of programming errors given in DS 6.6.3. Construct a point estimate of the average number of errors per month. What is the standard error of your point estimate?
Consider the data set of osteoporosis patient heights given in DS 6.6.4. Construct a point estimate of the average height. What is the standard error of your point estimate?
A sample of 31 data observations has a sample mean x- = 53.42 and a sample standard deviation S = 3.05. Construct a 95% two-sided t-interval for the population mean. (This problem is continued in Problem 8.1.9.)
Consider the sample of 41 glass sheets discussed in Problem 8.1.2. How many additional glass sheets should be sampled to construct a 9995 two-sided t-interval for the average sheet thickness with a length no larger than L0 = 0.05 mm?
Consider the sample of 20 breaking strength measurements discussed in Problem 8.1.3. How many additional data observations should be obtained to construct a 999? two-sided t-interval for the average breaking strength with a length no larger than L0 = 10.0?
A sample of 30 data observations has a sample mean x- = 14.62 and a sample standard deviation s = 2.98. bind the value of C for which μ ∈ (c. ∞) is a one sided 95% t-interval for the population mean μ. Is it plausible that μ > 16?
A sample of 61 bottles of chemical solution is obtained and the solution densities are measured. The sample mean is x- = 0.768 and the sample standard deviation is s = 0.0231. Find the value of c for which μ ∈ (c, ∞) is a one-sided 99% t-interval for the average solution density μ. Is it
A sample of 19 data observations has a sample mean of x- = 11.80. If an experimenter wishes to use a "known" value a = 2.0 for the population standard deviation, find the value of c for which μ ∈ (c, ∞) is a one-sided 95% confidence interval for the population mean μ.
A sample of 29 measurements of radiation levels in a research laboratory taken at random times has a sample mean of x- = 415.7. If an experimenter wishes to use a "known" value a = 10.0 for the standard deviation of these radiation levels based upon prior experience, find the value of C for which
The pH levels of a random sample of 16 chemical mixtures from a process were measured, and a sample mean x- = 6.861 and a sample standard deviation s = 0.440 were obtained. The scientists presented a confidence interval (6.668. 7.054) for the average pH level of chemical mixtures from the process.
Chilled cast iron is used for mechanical components that need particularly high levels of hardness and durability. In an experiment to investigate the corrosion properties of a particular type of chilled cast iron, a collection of x = 10 samples of this chilled cast iron provided corrosion rates
The data set of service times given in DS 6.1.4.
The data set of calls received by a switchboard given in DS 6.1.6.
A random sample of 41 glass sheets is obtained and their thicknesses are measured. The sample mean is x- = 3.04 mm and the sample standard deviation is v = 0.124 mm. Construct a 99% two-sided t-interval for the mean glass thickness. Do you think it is plausible that the mean glass thickness is 2.90
The data set of paving slab weights given in DS 6.1.7.
The data set of paint thicknesses given in DS 6.1.8.
The data set of plastic panel bending capabilities given in DS 6.1.9.
The yields of nine batches of a chemical process were measured and a sample mean of 2.843 and a sample standard deviation of 0.150 were obtained. The experimenter presented a confidence interval of (2.773.∞) for the average yield of the process. What is the confidence level of this confidence
Consider the data set 34 45 27 33 38 41 45 29 30 39 34 40 28 33 36 (a) What is the sample median? (b) Construct a 99% two-sided confidence interval for the population mean.
A random sample of 14 chemical solutions is obtained, and their strengths are measured. The sample mean is 5437.2 and the sample standard deviation is 376.9. (a) Construct a two-sided 95% confidence interval for the average strength. (b) Estimate how many additional chemical solutions need to be
A boot manufacturer is testing the quality of leather provided by a potential supplier. The manufacturer wants to construct a two-sided confidence interval with a confidence level of 959? that has a length no larger than 0.1. and from previous experience it is believed that the variability in the
The breaking strengths of a random sample of 20 bundles of wool fibers have a sample mean x- = 436.5 and a sample standard deviation s = 11.90. Construct 90%, 95%, and 99% two-sided t-intervals for the average breaking strength. Compare the lengths of the confidence intervals. Do you think it is
A random sample of 16 one-kilogram sugar packets is obtained and the actual weights of the packets are measured. The sample mean is x- = 1.053 kg and the sample standard deviation is s =,0.058 kg. Construct a 99% two-sided t-interval for the average sugar packet weight. Do you think it is plausible
A sample of 28 data observations has a sample mean x = 0.0328. If an experimenter wishes to use a "known" value a = 0.015 for the population standard deviation, construct an appropriate 95% two-sided confidence interval for the population mean
The resilient moduli of 10 samples of a clay mixture are measured and the sample mean is -x = 19.50. If an experimenter wishes to use a "known" value σ = 1.0 for the standard deviation of the resilient modulus measurements based upon prior experience, construct appropriate 90%, 95%, and 99%
An experimenter feels that a population standard deviation is no larger than 10.0 and would like to construct a 95% two-sided t-interval for the population mean that has a length at most 5.0. What sample size would you recommend.'
An experimenter would like to construct a 99% two side0d t-interval, with a length at most 0.2 ohms, for the average resistance of a segment of copper cable of a certain length. If the experimenter feels that the standard deviation of such resistances is no larger than 0.15 ohms, what sample size
Consider the sample of 31 data observations discussed in Problem 8.1.1. How man) additional data observations should be obtained to construct a 9595 two-sided μ-interval for the population mean n with a length no larger than L0 = 2.0?
A sample of n = 18 observations has a sample mean of x- = 57.74 and a sample standard deviation of s = 11.20. Consider the hypothesis testing problems (a) H0: μ = 55.0 versus HA: p ≠ 55.0 (b) H0: μ > 65.0 versus HA: p < 65.0 In each case, write down an expression for the p-value. What do the
An experimenter is interested in the hypothesis testing problem H0 : μ > 420.0 versus HA : μ < 420.0 where p is the average radiation level in a research laboratory. Suppose that a sample of n = 29 radiation level measurements is obtained and that the experimenter wishes to use a value of σ
A machine is set to cut metal plates to a length of 44.350 mm. The lengths of a random sample of 24 metal plates have a sample mean of x- = 44.364 mm and a sample standard deviation of .v = 0.019 mm. Is there any evidence that the machine is miscalibrated?
A food manufacturer claims that at the time of purchase by a consumer the average age of its product is no more than 120 days. In an experiment to test this claim a random sample of 36 items are found to have ages at the time of purchase with a sample mean of x- = 122.5 days and a sample standard
A chemical plant is required to maintain ambient sulfur levels in the working environment atmosphere at an average level of no more than 12.50. The results of 15 randomly timed measurements of the sulfur level produced a sample mean of x- = 14.82% and a sample standard deviation of s = 2.91%. What
A company advertises that its electric motors provide an efficiency that is at least 25% higher than the industry norm. A consumer interest group ran an experiment with a sample of 23 machines for which the increases in efficiency over the industry norm had a sample mean of x- = 22.8% and a sample
Consider the data set of calls received by a switchboard given in DS 6.1.6. A manager claims that the switchboard needs additional staffing because the average number of calls taken per minute is at least 13. How do you-feel about this claim?
Consider the data set of paving slab weights given in DS 6.1.7. The slabs are supposed to have an average weight of 1.1 kg. Is there any evidence that the manufacturing process needs adjusting?
Consider the data set of calls received by a switchboard given in DS 6.1.6. A manager claims that the switchboard needs additional staffing because the average number of calls taken per minute is at least 13. How do you-feel about this claim? Discuss.
Consider the data set of paving slab weights given in DS 6.1.7. The slabs are supposed to have an average weight of 1.1 kg. Is there any evidence that the manufacturing process needs adjusting? Discuss.
Consider the data set of paint thicknesses given in DS 6.1.8. The spray painting machine is supposed to spray paint to a mean thickness of 0.225 mm. What is the evidence that the spray painting machine is not performing properly?
A sample of n = 39 observations has a sample mean of .x- = 5532 and a sample standard deviation of s = 287.8. Consider the hypothesis testing problems (a) H0: μ = 5680 versus HA: μ ≠ 5680 (b) H0: μ < 5450 versus HA: μ > 5450 In each case, write down an expression for the p-value. What do the
Consider the data set of plastic panel bending capabilities given in DS 6.1.9. The plastic panels are designed to be able to bend on average to at least 9.5 without • deforming. Is there any evidence that this design criterion has not been met?
An experimenter randomly selects n = 16 batteries from a production line and measures their voltages. An average x- = 239.13 is obtained, with a sample standard deviation s = 2.80. Does this experiment provide sufficient evidence for the experimenter to conclude that the average voltage of the
A two-sided t-procedure is performed. Use Table III to put bounds on the p-value if: (a) n = 12, t = 3.21 (b) n = 24, t = 1.96 (c) n = 30, t = 3.88
A company claims that its components have an average length of 82.50 mm. An experimenter tested this claim by measuring the lengths of a random sample of 25 components. It was found that x- = 82.40 and s = 0.14. Use a hypothesis test to assess whether the experimenter has sufficient evidence to
A random sample of 25 components is obtained, anil then-weights are measured. The sample mean is 71.97 g and the sample standard deviation is 7.44 g. Conduct a hypothesis test to assess whether there is sufficient evidence to establish that the components have an average weight larger than 70 g.
A random sample of 28 plastic items is obtained, and their breaking strengths are measured. The sample mean is 7.442 and the sample standard deviation is 0.672. Conduct a hypothesis test to assess whether there is any evidence that the average breaking strength is not 7.000.
An experimenter measures the failure times of a random sample of 25 components. The sample average is 53.43 hours and the sample standard deviation is 3.93 hours. Use a hypothesis test to determine whether there is sufficient evidence for the experimenter to conclude that the average failure time
Use Table III to indicate whether the p-values for the following t-tests are less than 1%, between 1% and 10%. or more than 10%. (a) H0: μ = 10, HA: μ ≠ 10, n = 20, x- = 12.49, s = 1.32 (b) H0 : μ < 3.2, HA : μ > 3.2, n = 43, x- = 3.03, s = 0.11 (c) H0: μ > 85. HA: μ < 85. h = 16, x- =
A sample of n = 13 observations has a sample mean of -x = 2.879. If an assumed known standard deviation of σ = 0.325 is used, calculate the p-values for the hypothesis testing problems (a) H0: n = 3.0 versus HA: μ ≠ 3.0 (b) H0: μ > 3.1 versus HA: μ < 3.1
A sample of n = 44 observations has a sample mean of x- = 87.90. If an assumed known standard deviation of CT = 5.90 is used, calculate the p-values for the hypothesis testing problems (a) H0: μ = 90.0 versus HA: μ ≠ 90.0 (b) H0: μ < 86.0 versus HA: n > 86.0
An experimenter is interested in the hypothesis testing problem H0: μ = 3.0 mm versus H : n ≠ 3.0 mm where μ is the average thickness of a set of glass sheets. Suppose that a sample of n = 41 glass sheets is obtained and their thicknesses are measured. (a) For what values of the
An experimenter is interested in the hypothesis testing problem H0: μ = 430.0 versus HA : μ ≠ 430.0 where p is the average breaking strength of a bundle of wool fibers. Suppose that a sample of n = 20 wool fiber bundles is obtained and their breaking strengths are measured. (a) For what
An experimenter is interested in the hypothesis testing problem H0: μ = 1.025 kg versus HA: μ ≠ 1.025 kg where μ is the average weight of a I-kilogram sugar packet. Suppose that a sample of n = 16 sugar packets is obtained and their weights are measured. (a) For what values of the
An experimenter is interested in the hypothesis testing problem H0: μ = 20.0 versus HA: μ ≠ 20.0 where p is the average resilient modulus of a clay mixture. Suppose that a sample of n = 10 resilient modulus measurements is obtained and that the experimenter wishes to use a value of σ = 1.0
An experimenter is interested in the hypothesis testing problem H0: μ < 0.065 versus HA: μ > 0.065 where μ is the average density of a chemical solution. Suppose that a sample of n = 61 bottles of the chemical solution is obtained and their densities are measured. (a) For what values of the
In an experiment to investigate when a radar picks up a certain kind of target, a total of n = 15 trials arc conducted in which the distance of the target from the radar is measured when the target is detected. A sample mean of x- = 67.42 miles is obtained, with a sample standard deviation of s =
A sample of n = 18 observations has a sample standard deviation of s = 6.48. Use the method above to construct 9995 and 95% two-sided confidence intervals for the population variance σ2.
Consider the data set of 41 glass sheet thicknesses described in Problem 8.1.2. Construct a 99% two-sided confidence interval for the standard deviation a of the sheet thicknesses.
Consider the data set of breaking strengths of wool fiber bundles described in Problem 8.1.3. Construct a 95% two-sided confidence interval for the variance a2 of the breaking strengths.
Consider the data set of sugar packet weights described in Problem 8.1.4. Construct 90%. 95%, and 99% two-sided confidence intervals for the standard deviation a of the packet weights.
A two-sided t - test is performed. Use Table III to put bounds on the p-value if: (a) n = 8, t = 1.31 (b) n = 30, t = -2.82 (c) n = 25. t = 1.92
An experimenter measures the compressibility of 16 samples of clay randomly selected from a particular location, and they have a sample mean of 76.99 and a sample standard deviation of 5.37. Does this provide sufficient evidence for the experimenter to conclude that the average clay compressibility
A sample of 14 fibers was tested. Their strengths had a sample average of 266.5 and a sample standard deviation of 18.6. Use a hypothesis test to assess whether it is sale to conclude that the average strength of fibers of this type is at least 260.0.
8.5.18 Consider the data set 34 54 73 38 89 52 75 33 50 39 42 42 40 66 72 85 28 71 which is a random sample from a distribution with an unknown mean p. Calculate the following. (a) The sample size. (b) The sample median, (c) The sample mean. (d) The sample standard deviation. (e) The sample
Are the following statements true or false? (a) In hypothesis testing the null hypothesis can never be proved to be correct. (b) For a given data set a two-sided confidence interval for a parameter with a confidence level 99% is shorter than a two-sided confidence interval for the parameter with a
A company is planning a large telephone survey and is interested in assessing how long it will take. In a short pilot study, 40 people are contacted by telephone and are asked the specified set of questions. The times of these 40 telephone surveys have a sample mean of x- = 9.39 minutes, with a
A sample of 22 wires was tested. Their resistances had a sample average of 193.7 and a sample standard deviation of 11.2. It is claimed that the average resistance of wires of this type is 200.0. Use an appropriate hypothesis test to investigate this claim.
An engineer selects 10 components at random and measures their strengths. It is reported that the average strength of the components is between 72.3 and 74.5 with 99% confidence. (a) What is the sample standard deviation of the 10 component strengths? (b) If a 99% two-sided confidence interval is
A random sample of 10 items gives x- = 614.5 and s = 42.9. (a) Use a hypothesis test to determine whether there is sufficient evidence for the experimenter to conclude that the population average is not 600. (b) Construct a 99% two-sided confidence interval for the population average. (c) If a 99%
Twelve samples of a metal alloy are tested. The flexibility measurements had a sample average of 732.9 and a sample standard deviation of 12.5. (a) Is there sufficient evidence to conclude that the flexibility of this kind of metal alloy is smaller than 750? Use an appropriate hypothesis test to
Flow meters are installed in urban sewer systems to measure the flows through the pipes. In dry weather conditions (no rain) the flows are generated by waste water from households and industries, together with some possible drainage from water stored in the topsoil from previous rainfalls. In a

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