Questions and Answers of Applied Statistics and Probability

Use the data from Exercise 8-59 to compute a two-sided Agresti-Coull CI on the proportion of defective circuits. Compare and discuss the CI to the one computed in Exercise 8-59.
Information on a packet of seeds claims that 93% of them will germinate. Of the 200 seeds that I planted, only 180 germinated.(a) Find a 95% confidence interval for the true proportion of seeds that
The U.S. Postal Service (USPS) has used optical character recognition (OCR) since the mid-1960s. In 1983, USPS began deploying the technology to major post offices throughout the country
An article in the Journal of the American Statistical Association (1990, Vol. 85, pp. 972–985) measured the weight of 30 rats under experiment controls. Suppose that 12 were underweight
Of 1000 randomly selected cases of lung cancer, 823 resulted in death within 10 years. (a) Calculate a 95% two-sided confidence interval on the death rate from lung cancer.(b) Using the point
The 2004 presidential election exit polls from the critical state of Ohio provided the following results. The exit polls had 2020 respondents, 768 of whom were college graduates. Of the college
An article in Knee Surgery, Sports Traumatology, Arthroscopy [“Arthroscopic Meniscal Repair with an Absorbable Screw: Results and Surgical Technique” (2005, Vol. 13, pp. 273–279)] showed that
The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 13 defectives. (a) Calculate a 95%
From the data on CAT scans in Exercise 8-45(a) Find a two-sided 95% confidence interval for the standard deviation.(b) What should you do to address any reservations about the confidence interval you
From the data on the pH of rain in Ingham County, Michigan:Find a two-sided 95% confidence interval for the standard deviation of pH. 5.47 5.37 5.38 4.63 5.37 3.74 3.71 4.96 4.64 5.11 5.65 5.39
An article in the Australian Journal of Agricultural Research [€œNon-Starch Polysaccharides and Broiler Performance on Diets Containing Soyabean Meal as the Sole Protein
An article in Technometrics (1999, Vol. 41, pp. 202€“ 211) studied the capability of a gauge by measuring the weight of paper. The data for repeated measurements of one sheet of paper are
An article in Cancer Research [€œAnalyses of Litter- Matched Time-to-Response Data, with Modifications for Recovery of Interlitter Information€ (1977, Vol. 37, pp.
An article in Urban Ecosystems, €œUrbanization and Warming of Phoenix (Arizona, USA): Impacts, Feedbacks and Mitigation€ (2002, Vol. 6, pp. 183€“203), mentions that
An article in Medicine and Science in Sports and Exercise [“Electro stimulation Training Effects on the Physical Performance of Ice Hockey Players” (2005, Vol. 37, pp. 455–460)] considered the
Consider the situation in Exercise 8-48. Find a 99% lower confidence bound on the standard deviation.
A healthcare provider monitors the number of CAT scans performed each month in each of its clinics. The most recent year of data for a particular clinic follows (the reported variable is the number
Using the data from Exercise 8-22 on adhesion without assuming that the standard deviation is known,(a) Check the assumption of normality by using a normal probability plot.(b) Find a 95% confidence
An article in Computers & Electrical Engineering [€œParallel Simulation of Cellular Neural Networks€ (1996, Vol. 22, pp. 61€“84)] considered the speedup of cellular
A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected, and the diameter is measured. The resulting data (in millimeters) are as follows:(a)
The compressive strength of concrete is being tested by a civil engineer who tests 12 specimens and obtains the following data:(a) Check the assumption that compressive strength is normally
The solar energy consumed (in trillion BTU) in the United States by year from 1989 to 2004 is shown in the following table. Read down then across for year.Check the assumption of normality in the
The Bureau of Meteorology of the Australian Government provided the mean annual rainfall (in millimeters) in Australia 1983€“2002 as follows (http://www.bom.gov.au/
An article in Obesity Research [“Impaired Pressure Natriuresis in Obese Youths” (2003, Vol. 11, pp. 745–751)] described a study in which all meals were provided for 14 lean boys for three days
An article in Medicine and Science in Sports and Exercise [“Maximal Leg-Strength Training Improves Cycling Economy in Previously Untrained Men” (2005, Vol. 37, pp. 131–136)] studied cycling
A random sample has been taken from a normal distribution. Output from a software package follows:(a) Fill in the missing quantities.(b) Find a 95% CI on the population mean. Variable N Mean SE Mean
A random sample has been taken from a normal distribution. Output from a software package follows:(a) Fill in the missing quantities.(b) Find a 95% CI on the population mean. Variable N Mean SE Mean
Dairy cows at large commercial farms often receive injections of bST (Bovine Somatotropin), a hormone used to spur milk production. Bauman et al. (Journal of Dairy Science, 1989) reported that 12
Ishikawa et al. (Journal of Bioscience and Bioengineering, 2012) studied the adhesion of various biofilms to solid surfaces for possible use in environmental technologies. Adhesion assay is conducted
An article in the Journal of Agricultural Science [€œThe Use of Residual Maximum Likelihood to Model Grain Quality Characteristics of Wheat with Variety, Climatic and Nitrogen Fertilizer
Following are two confidence interval estimates of the mean m of the cycles to failure of an automotive door latch mechanism (the test was conducted at an elevated stress level to accelerate the
Consider the gain estimation problem in Exercise 8-4.(a) How large must n be if the length of the 95% CI is to be 40?(b) How large must n be if the length of the 99% CI is to be 40?
A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (37.53, 49.87) and (35.59, 51.81)(a) What is the value of the sample
A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.02, 61.98) and (39.95, 60.05)(a) What is the value of the sample
Consider the one-sided confidence interval expressions for a mean of a normal population.(a) What value of zα would result in a 90% CI?(b) What value of zα would result in a 95% CI?(c) What value
For a normal population with known variance σ2, answer the following questions: (a) What is the confidence level for the interval x̅ − 2.14σ √n ≤ μ ≤ x̅ + 2.14σ √n ?(b) What is
Order Statistics. Let X1, X2, €¦. Xnbe a random sample of size n from X, a random variable having distribution function F(x). Rank the elements in order of increasing numerical magnitude,
Consistent Estimator. Another way to measure the closeness of an estimator Î˜ to the parameter Î¸ is in terms of consistency. If Î˜nis an estimator of Î¸
An electric utility has placed special meters on 10 houses in a subdivision that measures the energy consumed (demand) at each hour of the day. The company is interested in the energy demand at one
You plan to use a rod to lay out a square, each side of which is the length of the rod. The length of the rod is μ, which is unknown. You are interested in estimating the area of the square, which
Let f x = (1 / θ)x (1- θ)/θ , 0 < x < 1, and < θ < ∞ show that Θ = −(1 / n)Σni=1 ln(X ) is the maximum likelihood estimator for θ and that Θ is an unbiased estimator for q.
Let f (x) = θxθ−1 , 0<θ< ∞, and 0< x <1. Show that Θ = − n / (ln Πni=1 Xi) is the maximum likelihood estimator for θ.
A manufacturer of semiconductor devices takes a random sample of 100 chips and tests them, classifying each chip as defective or nondefective. Let Xi= 0 if the chip is nondefective and Xi= 1 if the
A random sample of size n = 16 is taken from a normal population with μ = 40 and σ2 = 5. Find the probability that the sample mean is less than or equal to 37.
The time between failures of a machine has an exponential distribution with parameter λ. Suppose that the prior distribution for λ is exponential with mean 100 hours. Two machines are observed, and
The weight of boxes of candy is a normal random variable with mean μ and variance 1/10 pound. The prior distribution for μ is normal with mean 5.03 pound and variance 1/ 25 pound. A random sample
Suppose that X is a normal random variable with unknown mean and known variance σ2 = 9. The prior distribution for μ is normal with μ0 = 4 and σ20 = 1. A random sample of n = 25 observations is
Suppose that X is a Poisson random variable with parameter λ. Let the prior distribution for λ be a gamma distribution with parameters m + 1 and (m +1) / λ0.(a) Find the posterior distribution for
Suppose that X is a normal random variable with unknown mean μ and known variance σ2. The prior distribution for μ is a uniform distribution defined over the interval [a, b]. (a) Find the
Suppose that X is a normal random variable with unknown mean μ and known variance σ2. The prior distribution for μ is a normal distribution with mean μ0 and variance σ20 . Show that the Bayes
Reconsider the oxide thickness data in Exercise 7-35 and suppose that it is reasonable to assume that oxide thickness is normally distributed. (a) Compute the maximum likelihood estimates of μ
Let X1, X2, €¦ , Xnbe uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is aÌ‚ = max(Xi).(a) Argue intuitively why Ë†a
The Rayleigh distribution has probability density function(a) It can be shown that E(X2) = 2Î¸. Use this information to construct an unbiased estimator for Î¸.(b) Find the
Consider the Poisson distribution with parameter λ. Find the maximum likelihood estimator of λ, based on a random sample of size n.
Suppose that two independent random samples (of size n1 and n2) from two normal distributions are available. Explain how you would estimate the standard error of the difference in sample means X̅1
Consider a normal random variable with mean 10 and standard deviation 4. Suppose that a random sample of size 16 is drawn from this distribution and the sample mean is computed. We know that the
An exponential distribution is known to have a mean of 10. You want to find the standard error of the median of this distribution if a random sample of size 8 is drawn. Use the bootstrap method to
Suppose that the random variable X has a lognormal distribution with parameters θ = 1.5 and ω = 0.8. A sample of size n = 15 is drawn from this distribution. Find the standard error of the sample
XÌ…1and S21are the sample mean and sample variance from a population with mean Î¼1 and variance Ïƒ12. Similarly, XÌ…2and S22are the sample mean and
Suppose that we have a random sample X1, X2, …., Xn from a population that is N(μ, σ2). We plan to use Θ = Σni=1 (Xi – X̅)2 / c to estimate σ2. Compute the bias in Θ as an
Let X1and X2be independent random variables with mean Î¼ and variance Ïƒ2. Suppose that we have two estimators of Î¼:(a) Are both estimators unbiased estimators of
A computer software package calculated some numerical summaries of a sample of data. The results are displayed here:(a) Fill in the missing quantities.(b) What is the estimate of the mean of the
A computer software package calculated some numerical summaries of a sample of data. The results are displayed here:(a) Fill in the missing quantities.(b) What is the estimate of the mean of the
Consider a Weibull distribution with shape parameter 1.5 and scale parameter 2.0. Generate a graph of the probability distribution. Does it look very much like a normal distribution? Construct a
Wayne Collier designed an experiment to measure the fuel efficiency of his family car under different tire pressures. For each run, he set the tire pressure and then measured the miles he drove on a
Like hurricanes and earthquakes, geomagnetic storms are natural hazards with possible severe impact on the Earth. Severe storms can cause communication and utility breakdowns, leading to possible
Researchers in the Hopkins Forest (see Exercise 7-16) also count the number of maple trees (genus acer) in plots throughout the forest. The following is a histogram of the number of live maples in
From the data in Exercise 6-21 on the pH of rain in Ingham County, Michigan:What proportion of the samples has pH below 5.0? 5.37 5.38 4.63 5.37 3.74 3.71 4.96 4.64 5.11 5.65 5.39 4.16 5.62 4.57 4.64
Scientists at the Hopkins Memorial Forest in western Massachusetts have been collecting meteorological and environmental data in the forest data for more than 100 years. In the past few years,
Consider the concrete specimens in Exercise 7-7. What is the standard error of the sample mean?
Consider the compressive strength data in Table 6-2. What proportion of the specimens exhibit compressive strength of at least 200 psi?
Consider the hospital emergency room data from Exercise 6-124. Estimate the proportion of patients who arrive at this emergency department experiencing chest pain.
Using the results of Exercise 6-130, which of the two quantities Σni=1 (xi – x̅)2 and Σni=1 (xi – μ)2 will be smaller, provided that x̅ ≠ μ?
Consider the quantity Σni=1 (xi – a)2. For what value of a is this quantity minimized?
Consider the global mean surface air temperature anomaly and the global CO2 concentration data originally shown in Table 6E.5.(a) Construct a scatter plot of the global mean surface air temperature
The force needed to remove the cap from a medicine bottle is an important feature of the product because requiring too much force may cause diffi culty for elderly patients or patients with arthritis
The energy consumption for 90 gas-heated homes during a winter heating season is given in Table 6E.16. The variable reported is BTU/number of heating degree days. (a) Calculate the sample mean
One of the authors (DCM) has a Mercedes-Benz 500 SL Roadster. It is a 2003 model and has fairly low mileage (currently 45,324 miles on the odometer). He is interested in learning how his car’s
Patients arriving at a hospital emergency department present a variety of symptoms and complaints. The following data were collected during one weekend night shift (11:00 p.m. to 7:00 a.m.):(a)
In their book Introduction to Time Series Analysis and Forecasting (Wiley, 2008), Montgomery, Jennings, and Kulahci presented the data on the drowning rate for children between one and four years old
Using the data on acid rain from Exercise 6-21,(a) Find the quartiles and median of the data.(b) Draw a box plot for the data.(c) Should any points be considered potential outliers? Compare this to
Using the data on bridge conditions from Exercise 6-20,(a) Find the quartiles and median of the data.(b) Draw a box plot for the data.(c) Should any points be considered potential outliers? Compare
Construct a frequency distribution and histogram for the swim time measurements in Exercise 6-24.
Construct a frequency distribution and histogram for the combined cloud-seeding rain measurements in Exercise 6-22.
Construct a frequency distribution and histogram for the acid rain measurements in Exercise 6-21.
Construct a frequency distribution and histogram for the bridge condition data in Exercise 6-20.
The female students in an undergraduate engineering core course at ASU self-reported their heights to the nearest inch. The data follow. Construct a stem-and-leaf diagram for the height data and
The net energy consumption (in billions of kilowatthours) for countries in Asia in 2003 was as follows (source: U.S. Department of Energy Web site, www.eia.doe.gov/emeu). Construct a stem-and-leaf
When will the median of a sample be equal to the mode?
When will the median of a sample be equal to the sample mean?
A back-to-back stem-and-leaf display on two data sets is conducted by hanging the data on both sides of the same stems. Here is a back-to-back stem-and-leaf display for the cloud seeding data in
For the data in Exercise 6-21,(a) Construct a stem-and-leaf diagram.(b) Many scientists consider rain with a pH below 5.3 to be acid rain (http://www.ec.gc.ca/eau-water/default.asp?
For the data in Exercise 6-20,(a) Construct a stem-and-leaf diagram.(b) Do any of the bridges appear to have unusually good or poor ratings?(c) If so, compute the mean with and without these bridges
In the 2000 Sydney Olympics, a special program initiated by IOC president Juan Antonio Samaranch allowed developing countries to send athletes to the Olympics without the usual qualifying procedure.
Construct dot diagrams of the seeded and unseeded clouds and compare their distributions in a couple of sentences.
Cloud seeding, a process in which chemicals such as silver iodide and frozen carbon dioxide are introduced by aircraft into clouds to promote rainfall was widely used in the 20th century. Recent
In an attempt to measure the effects of acid rain, researchers measured the pH (7 is neutral and values below 7 are acidic) of water collected from rain in Ingham County, Michigan.(a) Find the sample
The United States has an aging infrastructure as witnessed by several recent disasters, including the I-35 bridge failure in Minnesota. Most states inspect their bridges regularly and report their
Exercise 6-11 describes data from an article in Human Factors on visual accommodation from an experiment involving a high-resolution CRT screen. Data from a second experiment using a
An article in the Journal of Physiology [€œResponse of Rat Muscle to Acute Resistance Exercise Defined by Transcriptional and Translational Profiling€ (2002, Vol. 545, pp.

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