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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

A college decides to liberalize its admission policy. As a first step, the admissions committee decides to exclude student applicants scoring below the 20th percentile on the reading SAT. Translate
What is the Z score for a score of 150?
What percentage of scores are above 150?
What percentage of scores fall between 85 and 150?
Explain what is meant by scoring in the 95th percentile? What is the corresponding score?
What percentage of seniors scored lower than 300 on the math SAT?
What proportion of the teams have retention rates below Team B?
Estimate the percentage of U.S. adults who were victims at the 90% confidence level. Provide an interpretation of the confidence interval.
If you are asked to select a proportionate stratified sample of size 30 from the classroom, stratified by class level (senior, junior, etc.), how many students from each group will there be in the
If instead you are to select a disproportionate sample of size 20 from the classroom, with equal numbers of students from each class level in the sample, how many freshmen will there be in the sample?
When taking a random sample from a very large population, how does the standard error of the mean change when 1./ the sample size is increased from 100 to 1,600?2./ the sample size is decreased from
The following table shows the number of active military personnel in 2009, by region (including the District of Columbia).
Assume that σ = 226.83 for the entire population of 50 states. Calculate and interpret the standard error. (Consider the formula for the standard error. Since we provided the population standard
Write a brief statement on the following: the standard error compared with the standard deviation of the population, the shape of the sampling distribution, and suggestions for reducing the standard
Explain the concepts of estimation, point estimates, confidence level, and confidence interval
Calculate and interpret confidence intervals for means
Describe the concept of risk and how to reduce it
Calculate and interpret confidence intervals for proportions
What is the probability that the student will be a freshman?
Imagine that you choose one random student from the classroom (perhaps by using a random number table). What is the probability that the student will be a junior?
What is the percentile rank of Team A’s eligibility rate?
Describe the aims of sampling and basic principles of probability
Explain the relationship between a sample and a population
Identify and apply different sampling designs
Apply the concept of the sampling distribution
Describe the central limit theorem
Explain which of the following is a statistic and which is a parameter.1./ The mean age of Americans from the 2010 decennial census 2./ The unemployment rate for the population of U.S. adults,
The mayor of your city has been talking about the need for a tax hike. The city’s newspaper uses letters sent to the editor to judge public opinion about this possible hike, reporting on their
A friend interviews every 10th shopper who passes by her as she stands outside one entrance of a major department store in a shopping mall. What type of sample is she selecting? How might you define
A political polling firm samples 50 potential voters from a list of registered voters in each county in a state to interview for an upcoming election. What type of sample is this? Do you have enough
Another political polling firm in the same state selects potential voters from the same list of registered voters with a very different method. First, they alphabetize the list of last names, then
A social scientist gathers a carefully chosen group of 20 people whom she has selected to represent a broad cross-section of the population in New York City. She interviews them in depth for a study
\(X\) is the number of bits in error in the next four bits transmitted. What is the expected value of the square of the number of bits in error? Now, \(h(X)=X^{2}\). Therefore,\[ \begin{aligned}
In Example 4.1, \(X\) is the current measured in milliamperes. What is the expected value of power when the resistance is 100 ohms?
Determine the probability mass function of \(X\) from the following cumulative distribution function:\[ F(x)=\left\{\begin{array}{lr} 0 & x. \]Figure 3.3 displays a plot of \(F(x)\). From the
Two new product designs are to be compared on the basis of revenue potential. Marketing believes that the revenue from design A can be predicted quite accurately to be \(\$ 3\) million. The revenue
The probability that a wafer contains a large particle of contamination is 0.01. If it is assumed that the wafers are independent, what is the probability that exactly 125 wafers need to be analyzed
Consider the time to recharge the flash. The probability that a camera passes the test is 0.8, and the cameras perform independently. What is the probability that the third failure is obtained in
Let the continuous random variable \(X\) denote the current measured in a thin copper wire in milliamperes. Assume that the range of \(X\) is \([4.9,5.1] \mathrm{mA}\), and assume that the
The time until a chemical reaction is complete (in milliseconds) is approximated by the cumulative distribution function\[ F(x)= \begin{cases}0 & x
Assume that the current measurements in a strip of wire follow a normal distribution with a mean of 10 milliamperes and a variance of 4 (milliamperes) \({ }^{2}\). What is the probability that a
Suppose that the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and a variance of 4 (milliamperes) \({ }^{2}\). What is the
Assume that in a digital communication channel, the number of bits received in error can be modeled by a binomial random variable, and assume that the probability that a bit is received in error is
Assume that the number of asbestos particles in a squared meter of dust on a surface follows a Poisson distribution with a mean of 1000 . If a squared meter of dust is analyzed, what is the
In a large corporate computer network, user log-ons to the system can be modeled as a Poisson process with a mean of 25 log-ons per hour. What is the probability that there are no log-ons in an
Let \(X\) denote the time between detections of a particle with a Geiger counter and assume that \(X\) has an exponential distribution with \(E(X)=1.4\) minutes. The probability that we detect a
The time to prepare a slide for high-throughput genomics is a Poisson process with a mean of two hours per slide. What is the probability that 10 slides require more than 25 hours to prepare?
A semiconductor product consists of three layers. Suppose that the variances in thickness of the first, second, and third layers are 25,40 , and 30 square nanometers, respectively, and the layer
Consider an experiment that selects a cell phone camera and records the recycle time of a flash (the time taken to ready the camera for another flash). The possible values for this time depend on the
Suppose that the recycle times of two cameras are recorded. The extension of the positive real line \(R\) is to take the sample space to be the positive quadrant of the plane\[ S=R^{+} \times R^{+} \]
Each message in a digital communication system is classified as to whether it is received within the time specified by the system design. If three messages are classified, use a tree diagram to
Consider the sample space \(S=\{y y, y n, n y, n n\}\) in Example 2.2. Suppose that the subset of outcomes for which at least one camera conforms is denoted as \(E_{1}\). Then,\[ E_{1}=\{y y, y n, n
As in Example 2.1, camera recycle times might use the sample space \(S=R^{+}\), the set of positive real numbers. Let\[ E_{1}=\{x \mid 10 \leq x
The following table summarizes visits to emergency departments at four hospitals in Arizona. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining
The design for a Website is to consist of four colors, three fonts, and three positions for an image. From the multiplication rule, \(4 \times 3 \times 3=36\) different designs are possible.
A printed circuit board has eight different locations in which a component can be placed. If four different components are to be placed on the board, how many different designs are possible?
A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. We denote a knee and hip surgery as \(k\) and \(h\), respectively. The number of possible sequences of
A printed circuit board has eight different locations in which a component can be placed. If five identical components are to be placed on the board, how many different designs are possible?
A bin of 50 manufactured parts contains 3 defective parts and 47 nondefective parts. A sample of 6 parts is selected without replacement. That is, each part can be selected only once, and the sample
Assume that \(30 \%\) of the laser diodes in a batch of 100 meet the minimum power requirements of a specific customer. If a laser diode is selected randomly, that is, each laser diode is equally
A random experiment can result in one of the outcomes \(\{a, b, c, d\}\) with probabilities \(0.1,0.3,0.5\), and 0.1 , respectively. Let \(A\) denote the event \(\{a, b\}, B\) the event \(\{b, c,
Consider the inspection described in Example 2.11. From a bin of 50 parts, 6 parts are selected randomly without replacement. The bin contains 3 defective parts and 47 nondefective parts. What is the
Table 2.1 lists the history of 940 wafers in a semiconductor manufacturing process. Suppose that 1 wafer is selected at random. Let \(H\) denote the event that the wafer contains high levels of
Here is a simple example of mutually exclusive events, which are used quite frequently. Let \(X\) denote the \(\mathrm{pH}\) of a sample. Consider the event that \(X\) is greater than 6.5 but less
Table 2.2 provides an example of 400 parts classified by surface flaws and as (functionally) defective. For this table, the conditional probabilities match those discussed previously in this section.
Again consider the 400 parts in Table 2.2. From this table,\[ P(D \mid F)=\frac{P(D \cap F)}{P(F)}=\frac{10}{400} / \frac{40}{400}=\frac{10}{40} \]Note that in this example all four of the following
The probability that the first stage of a numerically controlled machining operation for high-rpm pistons meets specifications is 0.90 . Failures are due to metal variations, fixture alignment,
Consider the contamination discussion at the start of this section. The information is summarized here. Probability of Failure Level of Probability of Contamination Level 0.1 High 0.2 0.005 Not high
Consider the inspection described in Example 2.11. Six parts are selected randomly from a bin of 50 parts, but assume that the selected part is replaced before the next one is selected. The bin
Consider the inspection described in Example 2.11. Six parts are selected randomly without replacement from a bin of 50 parts. The bin contains 3 defective parts and 47 nondefective parts. Let \(A\)
Reconsider Example 2.20. The conditional probability that a high level of contamination was present when a failure occurred is to be determined. The information from Example 2.20 is summarized
Let \(D\) denote the event that you have the illness, and let \(S\) denote the event that the test signals positive. The probability requested can be denoted as \(P(D \mid S)\). The probability that
The simplest sequential description of the Ewens sampling formula is called the Chinese restaurant process. The first customer arrives and is seated at a table. After \(n\) customers have been
According to the standard definition in Sect. 11.4.2, two observational units \(u, u^{\prime}\) belong to the same experimental unit if the treatment assignment probabilities given the baseline
According to Villa et al.,Pigeons combat feather lice by removing them with their beaks during regular bouts of preening. Columbicola columbae, a parasite of feral pigeons, avoids preening by hiding
Explain why (5.3) is in conflict with randomization.
Use regress (. . . ) to compute the REML estimate of the variance components in (5.4). Hence obtain the estimated slopes, their difference, and the standard errors for all three.
Regress the \(32 \times 9\) lineage-time averages (for PC1) against sample size using sample size as weights. You should find a statistically significant positive coefficient a little larger than
Given the variance components, the Bayes estimate of the secular trend is a linear combination of the fitted mean vector and the fitted residual\[\tilde{\mu}=P Y+L \Sigma^{-1} Q Y \text {, }\]where
For the cubic and quadratic models described in the preceding exercise, compute the predicted temperature for next year, i.e., the conditional distribution of the mean temperature for next year given
This exercise is concerned with two versions of the Bayes estimate of the seasonal rainfall component, where it is required to compute \(E\left(\eta_{m} \midight.\) data) for each of 12 months. As
For the Oxford rainfall data up to Dec 2019, the first Bayes estimate in the preceding exercise is a flat \(10 \%\) shrinkage of monthly averages towards the annual average; the second Bayes estimate
In this exercise, \(\chi\) is the chordal metric on the unit circle. From the results of the preceding exercise, show that \(4 / \pi-\chi\left(t, t^{\prime}ight)\) is positive definite on \([0,2
Show that the quadratic function\[\frac{2 \pi^{2}}{3}-x(2 \pi-x)\]on \([0,2 \pi)\) has Fourier cosine coefficients \(4 \pi / k^{2}\) for \(k \geq 1\). Hence or otherwise, investigate the
Suppose that \(\eta \sim \operatorname{GP}(0, K)\), with \(K\) as defined in the preceding exercise. The tied-down process \(\zeta(t)=\eta(t)-\eta(0)\) is periodic and zero at integer multiples of
If you used the function \(g l m(y \sim r x 2\), family=Gamma(link= identity)) in the preceding exercise, you may have experienced a failure to converge. Write your own Newton-Raphson function with
From the cosine integral \(\int \cos (\omega t) e^{-|\omega|^{\alpha}} d \omega\), deduce that the \(\alpha\)-stable density has a Taylor series at the origin which begins\[\log p(t ; 1 / 2)=\text {
Check that the authors' lmer() code for the bee infection experiment produces the output shown in Sect. 9.2.2. Check that the revised lmer () code produces the output shown in Sect. 9.2.3.
The analysis of bee infection rates in Sect. 9.3 is only one of many similar analyses reported by Adler et al. (2020). A subsequent analysis examines how the mean infection intensity per colony is
All of the analyses of infection rates in Sect. 9.2 are for infections among surviving bumblebees, with the implicit assumption that the survival distribution does not depend on infection status.
Let \(a\) be the vector of bird ages, and \(n\) the vector of sequence lengths so that \(\check{a}=n+1-a\) is reverse age. Show that \(\operatorname{span}\{\mathbf{1}, a /(n+1)\}\) is equal to
Show that the model (10.1) implies exchangeability of terminal values \(Y_{i, n_{i}} \sim\) \(Y_{j, n_{j}}\) for every pair of birds, regardless of whether \(n_{i}=n_{j}\) and regardless of the years
In the light of the preceding exercises, discuss the pros and cons of using normalized versus unnormalized age in the mean model (10.1).
Show that the extended model suggested in the last paragraph of Sect. 10.2 also has exchangeable initial values and exchangeable terminal values. Under what conditions on the parameter do initial
Discuss the connection between sampling consistency as described in Sect. 11.6.1 and lack of interference as described in Sect. 11.4.8.
According to the discussion of Dong et al. (2015) in Sect. 12.1, the real-estate market in Beijing is divided up into 1117 land parcels, which are partitioned into 111 districts. These administrative
A different version of the preceding model uses the complementary log-log link function:\[\log \left(-\log \left(1-\operatorname{pr}\left(Y_{i, j} \leq
BIC is a sub-model selection procedure. For an observation y∈Rny∈Rn, BIC adds the penalty −12dlogn−12dlog⁡n to the maximized loglog likelihood for sub-models of dimension dd. Deduce from
Let \(Y_{1}, \ldots, Y_{n}\) be independent and identically distributed random variables whose distribution on \(\mathbb{R}\) is atom-free, and let \(r_{i}=\operatorname{rank}\left(Y_{i}ight)
Let \(Y_{1}, \ldots, Y_{n}\) be independent \(N\left(\beta, \sigma^{2}ight)\) random variables with parameter \((\beta, \sigma)\) taking values in \(\mathbb{R} \times \mathbb{R}^{+}\), and let the

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