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business
business statistics

Questions and Answers of Business Statistics

You have studied the salmon in the river Loppa, and are more interested in the weight variations than in the mean weight. The salmon weight follows a Normal distribution \(\phi_{(\mu, \sigma)}(x)\),
You have studied the brightness of a certain type of star, and are interested in the variation. The brightness follows a Normal distribution \(\phi_{(\mu, \sigma)}(x)\), so you study the variance by
Find \(t_{4,0.1}\).
In the following problems, the probability distributions for the parameter \(p\) of a Bernoulli process are given. Find the probability of event \(H\).(a) \(p \sim \beta_{(12,17)}(p) .
Let \(X\) be the proportion of \(40 \mathrm{~W}\) light bulbs that break when dropped onto a carpet from a height of \(1 \mathrm{~m}\). You have tried it out, and your probability distribution for
It is election time, and you are fact checking candidate April Weatherstone's factual claims. Let \(Y\) be the proportion of errors in Weatherstone's factual claims. Your estimate of \(Y\), after
T ∼ weib(5,4). What are μT, σ2T, and P(T ≤ 4)?
When Jack goes into the Canadian wilderness, he looks most forward to seeing reindeer. Let \(T\) be the time (in hours) that it takes before he sees a reindeer. \(T \sim\) weib \(_{(2,2)}\).(a) What
It is given that \(T\), the time (in seconds) it takes Santa's engineer elves to make a remote controlled car, is Weibull distributed with parameters \(\lambda=2.5\) and \(k=1\).(a) What is the
The longevity of Solan's motorized vehicles (in years), is \(T \sim f(x)=\) weib \(_{(1,0.5)}(x)\).(a) What are \(\mu_{T}\) and \(\sigma_{T}^{2}\) ?(b) What is the probability that a given motorized
X follows a continuous uniform probability distribution over the interval \(I=(a, b)\) if \(P(X=x)=1 /(b-a)\) whenever \(x \in I\), and 0 otherwise. Use the rules from sections 7.3, 7.5 and 7.6 to
X is (continuously) uniformly distributed over an interval \([a, b]\), and \(M \subset[a, b]\) is a disjoint union of intervals whose widths sum up to \(w\). What is \(P(X \in M)\) ?
Bayesian/frequentist(a) Who makes their estimates of the model parameters from the data alone?(b) Who speaks of \(P\) (observation | model )?(c) Who speaks of \(P\) (model | observation \()\) ?(d)
What are the two main purposes of statistical inference mentioned by Bard and Frederick? What is the difference between these two purposes, and how are the purposes related?
Your company has acquired the Chuck Wood's lumber mill. Along with the mill itself, they also got the mill's inventory. Your job is to estimate the humidity of the lumber by measuring 100 units.
Discuss strengths and weaknesses in Bard's and Frederick's estimates of the proportion of hugs Mina will give to each of them. May one of the ways of analysing fit better in one context, and the
Santa's workshop makes 10 different types of sack for Santa and his elves, types \(A_{1}, A_{2}, \ldots, A_{10}\). The number of \(A_{x}\) type sacks made are \(17 \times x\). In other words, there
You have \(100 A_{k}\), numbered from 1 to 100 , and \(f_{\text {pre }}(k)=k^{2} / 338350\). You make observation \(B_{1}\), and see that \(g(k)=P\left(B_{1} \mid A_{k}ight)=1 / k\). What are the
You participate in a chocolate lottery, where every participant gets a bag. Your Christmas elf has filled the bags like this: he puts a dark chocolate into the bag. Then he tosses a coin. If the coin
You have found a magic lamp on the beach. The genie in the lamp is a mathematician, and she has decided to fill a bag with rocks in the following way: she tossed a fair coin repeatedly until she got
Bernoulli trials- The alternatives are indexed by an \(x\) running from 1 through 15 .- The prior probability in choosing among the 15 alternatives is uniform.- For alternative \(x\), we have
Sacks with handles; the contents are white (W) and black (B) balls.- You have seven sacks with an index \(x\) running from 1 through 7.- Sack \(x\) has \(x\) handles, and in a random pick, each
Nuts come in two different chiralities (threadings); left-handed (L) and right-handed (R). Oleson's hardware store sells packs containing both kinds in one pack.- Oleson sells 5 types of nut pack.
You are updating a prior probabilityFind the posterior probability distribution \(f_{\text {post }}(x)\). fpre (x)= 0.5 +0.25x x = (-2,0] - 0.5 0.25x x = (0,2] 0 otherwise
Make a die or any other physical object that may serve as your "random generator". Divide the possible outcomes into two roughly equal sets. If, for instance, you have made a 6-sided die, you may
Prove the formula for the probability distribution of \(X_{+}\)in Section 13.1.1. Use that \(\mu\) and \(\left(X_{+}-\muight) \sim \phi_{(0, \sigma)}\) are independent.
Capacitors: You have measured the capacitance of FR Electronics's smallest capacitors. From a sample of 25 measurements, you got an average of \(\bar{c}=49.19 \mu \mathrm{F}\), and a sample standard
The tensile strength of cables of the same type and thickness typically follows some Normal distribution \(\phi_{(\mu, \sigma)}\), where \(\mu\) and \(\sigma\) depend only on type and thickness. A
You are given the prior hyperparameters of a binomial process, observation data, and a value \(p\). Find the posterior distribution for \(\pi\) and its Normal approximation. Further, calculate the
You are given the posterior for the Bernoulli parameter \(\pi\), and numbers \(m, s, k\), and \(l\). Find the predictive distributions for \(K_{+m}\) and \(L_{+s}\), and calculate the probabilities
The Bayern Munich player Arjen Robben scores most of his goals with the left foot. The statistics of the goals he has scored by foot are as follows:Let \(\pi\) be the proportion of Robben's foot
Bard has given you a biased coin, and you wonder what the probability \(\pi\) of heads is. He replies that he doesn't quite know, but that his friend Sam, who gave it to him, once estimated the
You are looking at the proportion \(\pi\) of consumers who prefer MegaCola to its competitors. You use Jeffreys' prior hyperparameters, \(a_{0}=b_{0}=0.5\).(a) You arrange blind tastings of MegaCola
You are estimating the proportion of Macintoshes among the laptops of a rather large company. Your prior hyperparameters are \(a_{0}=7\) and \(b_{0}=3\). You ask 10 laptop-using colleagues; 8 of them
You are looking into the quality of the diamonds of the diamond mines in a new area. You have a special interest in "Fancy diamonds" 5 of quality IF and VVS, and you are estimating \(\pi\), the
In the two-player board game Go, black and white take turns putting a stone on the board, with black having the first move. Sondre Glimsdal, the \(O g\) Go club chairman, wonders what percentage of
You are given prior hyperparameters for a Poisson process, and observational data. Find the posterior distribution for the rate parameter \(\lambda\).(a) Prior: \(\kappa_{0}=0, \tau_{0}=0\).
In the problems below, you are given the probability distribution of a stochastic variable \(X\) and a utility function \(u(x)\). Find the expected utility \(U\). (a) X ~ B(17.9) and u(x) = 3x + 2;
You are going to decide whether \(A\) : \(\Theta\theta_{0}\). The gain in utility of choosing \(A\) instead of \(B\) isIn the first three subproblems below, you are given \(\theta_{0}, w_{A}\), and
You are given the posterior distribution \(\theta \sim f(x)\), the significance \(\alpha\), and alternative hypothesis \(H_{1}\). Test, and decide between the competing hypotheses.(a) \(\theta \sim
You have a job controlling how well pubs fill pint servings. More precisely, you sample to evaluate if the mean servings \(\mu\) are at least 1.0 pint. For your job, you use a neutral prior. At one
Measuring 25 of FR Electronics's smallest capacitors, you got \(\bar{c}=49.19 \mu \mathrm{F}\) and sample standard deviation \(s_{c}=2.15 \mu \mathrm{F}\). Assume the capacitances follow a Normal
You are given a (posterior) distribution for \(\pi \sim \beta_{(a, b)}\), a significance \(\alpha\), and \(H_{1}\). Test the following competing hypotheses, to decide between them, both by direct
You are estimating \(\pi\), the proportion who prefer MegaCola to its competitors, and your posterior hyperparameters for \(\pi\) are \(a_{1}=41.5\) and \(b_{1}=9.5\). If the proportion who prefer
Your are estimating \(\pi\), the proportion of "Fancy diamonds" of quality IF and VVS, in a diamond mining project where they are considering buying new and expensive mining equipment if this
You are given a probability distribution for \(\tau \sim \gamma_{(k, \lambda)}\), a significance \(\alpha\), and \(H_{1}\). Determine the hypothesis test, both by direct calculation and by using the
The brothers Odd and Kjell Aukrust lie home in bed with whooping cough, and as Kjell rattles off a particularly long-lasting cough, Odd exclaims: That one lasted for rather a long time, but not as
Nicholas believes that the tomcat Baggins purrs for longer than the female cat Perry, but Caroline, who is a student keenly interested in statistics, asks him to back up his claim by hypothesis
Your are comparing two \(\gamma\) distributed variables \(\Theta \sim \gamma_{(k, l)}\) and \(\Psi \sim \gamma_{(m, n)}\) to decide between hypotheses \(H_{1}\) (as specified below) and \(H_{0}\),
You are comparing two \(\beta\) distributed variables \(\psi \sim \beta_{(k, l)}\) and \(\pi \sim \beta_{(m, n)}\) to determine the hypothesis \(H_{1}\) (which is either \(\psi>\pi\), or
Your company has for a long time used Imperial Deliveries for freight. Lately, however, a promising new competitor has surfaced: Centurium Falcon Freight. You decide to test the rate of delivery
From distribution to interval.(a) \(\mu \sim \phi_{(14,3)}\). Find \(I_{0.025, l}^{\mu}\) and \(I_{0.025, r}^{\mu}\) and \(I_{0.05}^{\mu}\).(b) \(\mu \sim \phi_{(-4.3,7.2)}\). Find \(I_{0.005,
From data + prior to interval. Find \(I_{2 \alpha}^{\mu}\) and \(I_{2 \alpha}^{+}\).(a) Data: \(\{0,1,2,3,4,5,6,7,8,9,10\}\), neutral prior, known \(\sigma=2.3 ; 2 \alpha=\) 0.05 .(b) Prior:
Sample size:(a) Given known \(\sigma=5\), and prior hyperparameter \(\kappa_{0}=0\), how many observations \(n\) do you have to make to ensure \(I_{0.01}^{\mu}\) is narrower than 0.5 ?(b) With known
You have tried weighing your dog, knowing well that it is unable to stand still on the scales. You have done this four times, and based on the wobbling of the weight dial, you assume the weighings
Your son is doing athletics, and his performance varies from day to day. He wants to compete, and has asked you to help him by assessing his high jumps. His coach is also a gymnastics and mathematics
From distribution to interval.(a) \(\mu \sim t_{(68.1,11.9,17)}\). Find \(I_{0.001}^{\mu}\).(b) \(\mu \sim t_{(68.1,11.9,4)}\). Find \(I_{0.001}^{\mu}\) and \(I_{0.1}^{\mu}\).(c) \(X_{+} \sim
From data + prior to interval. Find \(I_{2 \alpha}^{\mu}\) and \(I_{2 \alpha}^{+}\). In addition, find \(I_{2 \alpha}^{\tau}\) and \(I_{2 \alpha}^{\sigma}\). (a) Data:
Sample size:(a) Using prior hyperparameters \(\kappa_{0}=7, v_{0}=6\), and \(S S_{0}=17\), how many observations \(n\) do you need to make in order for \(P\left(\mu \in H_{1.3}^{\mu}ight) \geq 0.9\)
A standard European football goal is \(732 \mathrm{~cm}\) wide and \(244 \mathrm{~cm}\) high, so if the goalkeeper is standing in the middle of the goal, he needs to be able to throw himself far
You are out diving with five friends, and stop to admire a school of goldfish. At that point, you are all at the same depth. Your ACME depth gauges respectively 33.1, 28.3, 29.0, 29.7, 33.2, and 30.9
You are following the band Pünk Flöyd, and just attended a concert with a rather permeating smell of sweet smoke. The police had conducted random checks of 10 audience members, and had in total
In the Norwegian box office hit film Il Tempo Gigante, it was said that on Reodor Felgen's first run of his racing car Il Tempo Gigante, the seismograph in Bergen registered it as an earthquake in
From distribution to interval.(a) \(\lambda \sim \gamma_{(4,17)}\). Find \(I_{0.05}^{\lambda}\).(b) \(\lambda \sim \gamma_{(7,128)}\). Find \(I_{0.001}^{\lambda}\).(c) \(\lambda \sim
From data + prior to interval. Find \(I_{2 \alpha}^{\mu}\) and \(I_{2 \alpha}^{+}\).(a) Prior: \(\kappa_{0}=3, \tau_{0}=5\). Observed: \(n=13\) occurrences during \(t=20\) units. \(2
Your prior hyperparameter is \(\kappa_{0}=5\). How many observations do you need to make for the relative interval width \(r\) for a \(80 \%\) credible interval to be less than 0.2 ?
The daily catch for grouse hunters in an area is considered to be Poisson distributed with rate \(\lambda\). One day, you talked to 23 hunters from a certain part of the Lowlands, and their total
The number of plumbing gaskets that need changing every week in an apartment complex is assumed to follow a Poisson process with rate \(\lambda\). You are estimating this need for an apartment
The number of cracks in the tarmac per kilometer of road is assumed to follow a Poisson process with rate \(\lambda\). Your job is to find this rate for a lesser highway, and you have found 13 cracks
How many four-leaf clovers are there per square meter in a field of leaf clovers? Assume that the occurrence follows a Poisson process with rate \(\lambda\). You look at three independent leaf clover
The number of bacterial colonies per cubic centimeter in a certain polluted lake is assumed to be Poisson distributed with parameter \(\lambda\). You sample 1 deciliter, and find 157 bacterial
From distribution to interval.(a) \(\pi \sim \beta_{(43,96)}\). Find \(I_{0.1}^{\pi}\), both by exact calculation of \(\beta\), and by using Normal approximation.(b) \(\pi \sim \beta_{(7,128)}\).
From data + prior to interval. Find \(I_{2 \alpha}^{\pi}\).(a) \(2 \alpha=0.07\). Prior: \(a_{0}=3, b_{0}=5\).Observed: \(k=13\) positives and \(l=20\) negatives.(b) \(2 \alpha=0.1\). Prior:
Sample size: \(\pi\) has prior hyperparameters \(a_{0}=12\) and \(b_{0}=31\). How many new observations do you need to make to ensure that \(I_{0.2}^{\pi}\) is narrower than 0.05 ?
You are studying Bard's biased coin. Your prior hyperparameters for \(\pi\), the probability of heads, is \(a_{0}=9\) and \(b_{0}=12\). Find the \(90 \%\) symmetric credible interval for \(\pi\)
Sondre Glimsdal wants to estimate \(\pi\), the proportion of \(G o^{1}\) games won by white. His prior is \(\pi \sim \beta_{(7,7)}\). Sondre updates his estimate by observing new games. Find the \(80
Find the \(P \%=(1-\alpha) 100 \%\) interval estimates \(\widehat{I_{\alpha}^{\mu}}\) and \(\widehat{I_{\alpha}^{+}} ; \sigma\) is known. (a) \(\sigma_{0}=2\). Data:
Find the \(P \%=(1-\alpha) 100 \%\) interval estimates \(\widehat{I_{\alpha}^{\mu}}, \widehat{I_{\alpha}^{\sigma}}\) (use that \(\sigma=1 / \sqrt{\tau}\) ) and \(\widehat{I_{\alpha}^{+}} ; \sigma\)
Find the \((1-\alpha) 100 \%\) confidence interval \(\hat{I_{\alpha}^{p}}\).(a) \(k=17\) positive and \(l=25\) negative. \(\alpha=0.05\).(b) You have heard that Coca and Pepsi have an equal share in
Determine the hypothesis test outcome about the parameter \(\pi\) for a Bernoulli process; significance \(\alpha\).(a) \(H_{1}: \pi>0.5 . \alpha=0.05\). Observations: \(k=8\) positive and \(l=6\)
Determine the hypothesis test outcome about the mean \(\mu\) for a Gaussian process; significance \(\alpha\).(a) \(H_{1}: \quad \mu eq 25 . \quad \alpha=0.05\). Statistics: \(n=27, S_{x}=715.333 .
It is claimed that the mean compression strength for a certain kind of steel beam exceeds 60000 psi, and you have decided to determine the test outcome of this alternative hypothesis with
Determine the hypothesis test outcome concerning the variance \(\sigma^{2}\) of a Gaussian process; significance \(\alpha\).(a) \(H_{1}: \sigma^{2}25 . \alpha=0.02\). We have 100 observations with
We look at the steel beams in again. This time, we are testing the variance, and the alternative hypothesis is \(H_{1}: \sigma eq 666\). Determine the hypothesis test outcome with \(\alpha=0.1\).
In the problems below, you are given observational data \(\left\{\left(x_{i}, y_{i}ight)ight\}\), and information about \(\sigma\). Find the ...- posterior distribution of \(\tau\).- posterior
In the problems below, you are given observational data \(\left\{\left(x_{i}, y_{i}ight)ight\}\), and information about \(\sigma\). Find the ...- posterior distribution of \(\tau\).- posterior
In the problems below, you are given observational data \(\left\{\left(x_{i}, y_{i}ight)ight\}\), and information about \(\sigma\). Find the ...- \(P \%=\left(1-\alpha_{1}ight) 100 \%\) credible
The Norway Cup is a European football youth cup and training camp that has arranged at Ekebergsletta in Oslo every year since 1972 (except for in 1976, when the organizers, Bækkelagets Sportsklub,
In American football, there was a scandal where the New England Patriots had inflated their balls rather poorly ahead of a decisive game against the Indianapolis Colts. Our local football coach
The spectral slope is an important characteristic of voices and musical instruments. The spectral slope is the slope \(b\) of the regression line between \(\ln x\) and \(y\), where \(x\) is the
Investigate whether there is a (linear) relation between the weekly salaries of the best UK football strikers and the number of goals they score, with significance \(\alpha=0.2\). We will be using
You are investigating the solubility of xylose in water inside a pumping system. The water is pressurized, so you get temperatures well in excess of \(200^{\circ} \mathrm{C}\). The variables are
The position number ω(letter) tells us where in the alphabet the given letter is located. For instance: ω(“b”) = 2. Find the mean, median and population standard deviation for the vowel
For the functions f (x) above that are continuous probability distributions:let X ∼ f (x), and find Mx, x, and P(X 5).
In the subproblems below, X is a mixed distribution with k components Xk, with respective weights wk. For each subproblem, find μX, Var(X), and P(X ≥ 0). (a) 2 components. W = 0.2, w = 0.8. = -3,
In the subproblems below, there is one discontinuity. Graph Fc, Fd, and FX, and find μX and Var(X). ~ (a) Continuous: wc = 0.7, X f(x) = 0.1 for 0 x 10. Discrete: wd = 0.3, and p4 = 1. 2 (b)
We have a mixed distribution with two discontinuities. The continous part is given by wc = 0.2, and Xc ∼ f (x) = 0.1 for 0 ≤ x ≤ 10. The discrete part is given by wd = 0.8, and the two
The artist Sandra has insured her concert tour. Insurance agents Beowulf have calculated with a 12% probability that the tour will be in the red, and that, in such a case, the loss will be
For Z = (X, Y) given by following table, find the marginal probabilities, P(X + Y = 4), and the correlation ρxy: X=1 X 2 X=3 - = Y = 1 0.05 0.05 0.3 Y = 2 0.05 0.25 0.05 Y = 3 0.2 0.05 0
Z = (X, Y) ∼ fZ(x, y) = 4/5 ( 2 − x − y3) for ∫x, y ∈ [0, 1]. Find the marginal probabilities fx and fy, the covariance σxy, and determine whether X and Y are independent.

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