Questions and Answers of Applied Statistics and Probability

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.
The following data are direct solar intensity measurements (watts/m2) on different days at a location in southern Spain: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878, 909, 918, 558, 768,
Suppose that you add 10 to all of the observations in a sample. How does this change the sample mean? How does it change the sample standard deviation?
Can the sample standard deviation be equal to zero? If so, give an example.
For any set of data values, is it possible for the sample standard deviation to be larger than the sample mean? If so, give an example.
Will the sample mean always be the most frequently occurring data value in the sample?
Will exactly half of the observations in a sample fall below the mean?
Will the sample mean always correspond to one of the observations in the sample?
Use the properties of moment generating functions to show that a sum of p independent normal random variables with means μi and variances σi2 for i = 1,2, ...., p has a normal distribution.
This exercise extends the hypergeometric distribution to multiple variables. Consider a population with N items of k different types. Assume that there are N1items of type 1, N2items of type
Suppose that the range of the continuous variables X and Y is 0 < x < a and 0 < y < b. Also suppose that the joint probability density function fXY(x, y) = g(x)h(y), where g(x) is a
Suppose that the joint probability function of the continuous random variables X and Y is constant on the rectangle 0 < x < a, 0 < y < b. Show that X and Y are independent.
Show that if X1, X2,…, Xp are independent random variables and Y = c1X1 + c2X2 + ... + cpXp, V (Y ) = c12V (X1) + c22V (X2) +…+ c2pV1 (Xp) You may assume that the random variables are continuous.
Show that if X1, X2,…, Xp are independent, continuous random variables, P(X1 ∈ A1, X2 ∈ A2,…, Xp ∈ Ap) = P(X1 ∈ A1)P(X2∈ A2) … P(Xp ∈ Ap) for any regions A1, A2,…, Ap in the
Use moment generating functions to determine the normalized power [E(X4)]1/4 from a cycling power meter when X has a normal distribution with mean 200 and standard deviation 20 Watts.
The intensity (mW/mm2) of a laser beam on a surface theoretically follows a bivariate normal distribution with maximum intensity at the center, equal variance σ in the x and y directions, and zero
The power in a DC circuit is P = I2 /R where I and R denote the current and resistance, respectively. Suppose that I is normally distributed with mean of 200 mA and standard deviation 0.2 mA and R is
Determine the value of c such that the function f(x,y) = cx2y for 0 < x < 3 and 0 < y < 2 satisfies the properties of a joint probability density function.Determine the following:(a) P(X
Suppose X has a lognormal distribution with parameters θ and ω. Determine the probability density function and the parameters values for Y = Xγ for a constant γ > 0. What is the name of this
A marketing company performed a risk analysis for a manufacturer of synthetic fibers and concluded that new competitors present no risk 13% of the time (due mostly to the diversity of fibers
An order of 15 printers contains 4 with a graphics- enhancement feature, 5 with extra memory, and 6 with both features. Four printers are selected at random, without replacement, from this set. Let
Ifdetermine E(X), E(Y), V(X), V(Y), and Ï by reorganizing the parameters in the joint probability density function. w (x, y) = 10.72 (x-1) - 1.6(x– 1)(y – 2) + (y – 2)}° | exp 1.2π
Suppose that X and Y have a bivariate normal distribution with σX = 4, σY = 1, /, μY = 4, and ρ = − 0.2. Draw a rough contour plot of the joint probability density function.
The joint distribution of the continuous random variables X, Y, and Z is constant over the region x2 + y2 ≤ 1, 0< z < 4. Determine the following:(a) P(X2 + Y2 ≤ 0.5) (b) P(X2 + Y2 ≤
To evaluate the technical support from a computer manufacturer, the number of rings before a call is answered by a service representative is tracked. Historically, 70% of the calls are answered in
The backoff torque required to remove bolts in a steel plate is rated as high, moderate, or low. Historically, the probability of a high, moderate, or low rating is 0.6, 0.3, or 0.1, respectively.
The percentage of people given an antirheumatoid medication who suffer severe, moderate, or minor side effects are 10, 20, and 70%, respectively. Assume that people react independently and that 20
Show that the following function satisfies the properties of a joint probability mass function:Determine the following:(a) P(X <0.5, Y <1.5) (b) P(X ‰¤ 1)(c) P(X
Suppose that Xi has a normal distribution with mean μi and variance σi 2 , i = 1, 2. Let X1 and X2 be independent.(a) Find the moment-generating function of Y = X1 + X1.(b) What is the distribution
Let X1, X2 X3,..., be independent exponential random variables with parameter λ.(a) Find the moment-generating function of Y = X1 + X2 + …+ Xr .(b) What is the distribution of the random variable
A random variable X has the gamma distribution0 F(x) = говr Aryle X>0 " style="" class="fr-fic fr-dib">(a) Show that the moment-generating function of X is0 F(x) = говr Aryle X>0 Ma()-(1-£)"
A random variable X has the exponential distributionShow that the moment-generating function of X is(b) Find the mean and variance of X. f(x) = le, x>0 10-(1-) Mx(t) =
The continuous uniform random variable X has density function(a) Show that the moment-generating function is(b) Use MX t ( ) to find the mean and variance of X. F(x) =, asiSB asx
A continuous random variable X has the following probability distribution: f (x) = 4xe−2x , x > 0(a) Find the moment-generating function for X.(b) Find the mean and variance of X.
The chi-squared random variable with k degrees of freedom has moment-generating function MX (t)=(1−2t)− k / 2. Suppose that X1 and X2 are independent chi-squared random variables with k1 and k2
A random variable X has the Poisson distribution(a) Show that the moment-generating function is(b) Use MX (t) to fi nd the mean and variance of the Poisson random variable. f(x) = х%3D0,1,... x!
A random variable X has the discrete uniform distribution(a) Show that the moment-generating function is(b) Use MX (t) to fi nd the mean and variance of X. =, x=1,2,.m f(x) т e (1-ет) Mx(t) =
Power meters enable cyclists to obtain power measurements nearly continuously. The meters also calculate the average power generated over a time interval. Professional riders can generate 6.6 watts
The computational time of a statistical analysis applied to a data set can sometimes increase with the square of N, the number of rows of data. Suppose that for a particular algorithm, the
Derive the probability density function for a lognormal random variable Y from the relationship that Y = exp(W) for a normal random variable W with mean θ and variance ω2.
An aircraft is flying at a constant altitude with velocity magnitude r1 (relative to the air) and angle θ1 (in a two-dimensional coordinate system). The magnitude and direction of the wind are r2
The random variable X has the probability distributionDetermine the probability distribution of Y = (X €“ 2)2. 0srS4 х S«(x)=.
Suppose that X has the probability distribution fX (x) = 1, 1 ≤ x ≤ 2 Determine the probability distribution of the random variable Y = eX.
The velocity of a particle in a gas is a random variable V with probability distribution fV (v) = av2 e–bv v > 0 where b is a constant that depends on the temperature of the gas and the mass of
A random variable X has the probability distribution fX (x) = e–x, x ≥ 0 Determine the probability distribution for the following:(a) Y = X2(b) Y = X ½(c) Y = ln X
Suppose that X has a uniform probability distribution fX (x) ,= 1, 0 ≤ x ≤ 1 Show that the probability distribution of the random variable Y = –2 X is chi-squared with two degrees of freedom.
Suppose that X is a continuous random variable with probability distribution (a) Determine the probability distribution of the random variable Y = 2X + 10.(b) Determine the expected value of Y.
Let X be a binomial random variable with p = 0.25 and n = 3. Determine the probability distribution of the random variable Y = X2.
Suppose that X is a random variable with probability distribution fX (x) = 1/4, x = 1 2 3 4 Determine the probability distribution of Y = 2X + 1.
Consider the perimeter of a part in Example 5-32. Let X1 and X2 denote the length and width of a part with standard deviations 0.1 and 0.2 centimeters, respectively. Suppose that the covariance
Weights of parts are normally distributed with variance σ2. Measurement error is normally distributed with mean 0 and variance 0.5σ2, independent of the part weights, and adds to the part weight.
An article in Knee Surgery Sports Traumatology, Arthroscopy [“Effect of Provider Volume on Resource Utilization for Surgical Procedures” (2005, Vol. 13, pp. 273–279)] showed a mean time of 129
Making handcrafted pottery generally takes two major steps: wheel throwing and fi ring. The time of wheel throwing and the time of firing are normally distributed random variables with means of 40
X and Y are independent, normal random variables with E(X) = 2, V (X) = 5, E(Y ) = 6, and V (Y ) = 8.Determine the following:(a) E(3X + 2Y ) (b) V (3X + 2Y )(c) P(3X + 2Y <18) (d) P(3X +
Suppose that X has a standard normal distribution. Let the conditional distribution of Y given X = x be normally distributed with mean E(Y | x) = 2x and variance V(Y | x) = 2x. Determine the
Patients given drug therapy either improve, remain the same, or degrade with probabilities 0.5, 0.4, 0.1, respectively. Suppose that 20 patients (assumed to be independent) are given the therapy. Let
In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Let X and Y denote the milliliters of acid and base needed for
Suppose that X and Y have a bivariate normal distribution with σX = 0.04, σY = 0.08, μX = 3.00, μY = 7.70, and ρ = 0.Determine the following:(a) P(2.95< X <3.05) (b) P(7.60(c)
Four electronic ovens that were dropped during shipment are inspected and classifi ed as containing either a major, a minor, or no defect. In the past, 60% of dropped ovens had a major defect, 30%
A Web site uses ads to route visitors to one of four landing pages. The probabilities for each landing page are equal. Consider 20 independent visitors and let the random variables W, X, Y , and Z
Based on the number of voids, a ferrite slab is classified as either high, medium, or low. Historically, 5% of the slabs are classified as high, 85% as medium, and 10% as low. A sample of 20 slabs is
Test results from an electronic circuit board indicate that 50% of board failures are caused by assembly defects, 30% by electrical components, and 20% by mechanical defects. Suppose that 10 boards
Determine the covariance and correlation for the lengths of the minor and major axes in Exercise 5-29.Exercise 5-29The lengths of the minor and major axes are used to summarize dust particles that

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