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business statistics

Questions and Answers of Business Statistics

A researcher measured heart rate \((x)\) and oxygen uptake \((y)\) for one person under varying exercise conditions. He wishes to determine if heart rate, which is easier to measure, can be used to
A researcher is investigating the relationship between yield of potatoes \((y)\) and level of fertilizer \((x\).\() She divides a eld into eight plots of equal\) size and applied fertilizer at a di
A researcher is investigating the relationship between fuel economy and driving speed. He makes six runs on a test track, each at a di erent speed, and measures the kilometers traveled on one liter
The Police Department is interested in determining the e ect of alcohol consumption on driving performance. Twelve male drivers of similar weight, age, and driving experience were randomly assigned
A study on air pollution in a major city measured the concentration of sulfur dioxide on 25 summer days. The measurements were:(a) Form a stem-and-leaf diagram of the sulfur dioxide measurements.(b)
Dutch elm disease is spread by bark beetles that breed in the diseased wood. A sample of 100 infected elms was obtained, and the number of bark beetles on each tree was counted. The data are
A manufacturer wants to determine whether the distance between two holes stamped into a metal part is meeting speci cations. A sample of 50 parts was taken, and the distance was measured to nearest
The manager of a government department is concerned about the efciency in which his department serves the public. Speci cally, he is concerned about the delay experienced by members of the public
A random sample of 50 families reported the dollar amount they had available as a liquid cash reserve. The data have been put in the following frequency table:(a) Construct a histogram of the
In this exercise we see how the default settings in for producing boxplots in Minitab and in \(\mathrm{R}\) can be misleading because they do not take the sample size into account. We will generate
Barker and McGhie (1984) collected 100 slugs from the species Limax maximus around Hamilton, New Zealand. They were preserved in a relaxed state, and their length in millimeters \((\mathrm{mm})\) and
There are two events \(A\) and \(B . P(A)=4\) and \(P(B)=5\). The events \(A\) and \(B\) are independent.(a) Find \(P(A)\).(b) Find \(P\left(\begin{array}{ll}A \quad B\end{array}\right)\).(c) Find
There are two events \(A\) and \(B . P(A)=5\) and \(P(B)=3\). The events \(A\) and \(B\) are independent.(a) Find \(P(A)\).(b) Find \(P\left(\begin{array}{ll}A & B\end{array}\right)\).(c) Find
There are two events \(A\) and \(B . P(A)=4\) and \(P(B)=4 . P\left(\begin{array}{ll}A & B\end{array}\right)=\) 24 .(a) Are \(A\) and \(B\) independent events? Explain why or why not.(b) Find
There are two events \(A\) and \(B . P(A)=7\) and \(P(B)=8 . P\left(\begin{array}{ll}A & B\end{array}\right)=1\).(a) Are \(A\) and \(B\) independent events? Explain why or why not.(b) Find
A single fair die is rolled. Let the event \(A\) be the face showing is even. Let the event \(B\) be the face showing is divisible by 3 .(a) List out the sample space of the experiment.(b) List the
Two fair dice, one red and one green, are rolled. Let the event \(A\) be the sum of the faces showing is equal to seven. Let the event \(B\) be the faces showing on the two dice are equal.(a) List
Two fair dice, one red and one green, are rolled. Let the event \(A\) be the sum of the faces showing is an even number. Let the event \(B\) be the sum of the faces showing is divisible by 3 .(a)
Two dice are rolled. The red die has been loaded. Its probabilities are \(P(1)=P(2)=P(3)=P(4)=\frac{1}{5}\) and \(P(5)=P(6)=\frac{1}{10}\). The green die is fair. Let the event \(A\) be the sum of
Suppose there is a medical diagnostic test for a disease. The sensitivity of the test is .95 . This means that if a person has the disease, the probability that the test gives a positive response is
Suppose there is a medical screening procedure for a specific cancer that has sensitivity \(=.90\), and specific city \(=.95\). Suppose the underlying rate of the cancer in the population is .001 .
In the game of blackjack, also known as twenty-one, the player and the dealer are dealt one card face-down and one card face-up. The object is to get as close as possible to the score 21, without
After the hand, the cards are discarded, and the next hand continues with the remaining cards in the deck. The player has had an opportunity to see some of the cards in the previous hand, those that
A discrete random variable \(Y\) has discrete distribution given in the following table:(a) Calculate \(P(1(b) Calculate \(\mathrm{E}[Y]\).(c) Calculate \(\operatorname{Var}[Y]\).(d) Let \(W=2 Y+3\).
A discrete random variable \(Y\) has discrete distribution given in the following table:(a) Calculate \(P(0(b) Calculate \(\mathrm{E}[Y]\).(c) Calculate
Let \(Y\) be \(\operatorname{binomial}(n=5=6)\).(a) Calculate the mean and variance by lling in the following table:i. \(\mathrm{E}[Y]=\)ii. \(\operatorname{Var}[Y]=\)(b) Calculate the mean and
Let \(Y\) be \(\operatorname{binomial}(n=4=3)\).(a) Calculate the mean and variance by lling in the following table:i. \(\mathrm{E}[Y]=\)ii. \(\operatorname{Var}[Y]=\)(b) Calculate the mean and
Suppose there is an urn containing 20 green balls and 30 red balls. A single trial consists of drawing a ball randomly from the urn, recording its color, and then putting it back in the urn. The
Suppose there is an urn containing 20 green balls and 30 red balls. A single trial consists of drawing a ball randomly from the urn, recording its color. This time the ball is not returned to the
Let \(Y\) have the Poisson \((=2)\) distribution.(a) Calculate \(P(Y=2)\).(b) Calculate \(P\left(\begin{array}{ll}Y & 2\end{array}\right)\).(c) Calculate \(P(1 \quad Y
Let \(Y\) have the Poisson \((=3)\) distribution.(a) Calculate \(P(Y=3)\).(b) Calculate \(P\left(\begin{array}{ll}Y & 3\end{array}\right)\).(c) Calculate \(P(1 \quad Y
Let \(X\) and \(Y\) be jointly distributed discrete random variables. Their joint probability distribution is given in the following table:(a) Calculate the marginal probability distribution of
There is an urn containing 9 balls, which can be either green or red. The number of red balls in the urn is not known. One ball is drawn at random from the urn, and its color is observed.(a) What is
Suppose we look at the two draws from the urn (without replacement) as a single experiment. The results were first draw red, second draw green. Find the posterior distribution of \(X\) by filling in
Let \(Y_{1}\) be the number of successes in \(n=10\) independent trials where each trial results in a success or failure, and , the probability of success, remains constant over all trials. Suppose
Suppose another 5 independent trials of the experiment are performed and \(Y_{2}=2\) successes are observed. Use the posterior distribution for from Exercise 6.4 as the prior distribution for . Find
Suppose we combine all the \(n=15\) trials all together and think of them as a single experiment where we observed a total of 9 successes. Start with the initial equally weighted prior from Exercise
Let \(Y\) be the number of counts of a Poisson random variable with mean . Suppose the 5 possible values of are \(1,2,3,4\), and 5 . We do not have any reason to give any possible value more weight
Let \(X\) have a \(\operatorname{beta}(35)\) distribution.(a) Find \(\mathrm{E}[X]\).(b) Find \(\operatorname{Var}[X]\).
Let \(X\) have a \(\operatorname{beta}(12\) 4) distribution.(a) Find \(\mathrm{E}[X]\).(b) Find \(\operatorname{Var}[X]\).
Let \(X\) have the uniform distribution.(a) Find \(\mathrm{E}[X]\).(b) Find \(\operatorname{Var}[X]\).(c) Find \(P\left(\begin{array}{ll}X \quad 25\end{array}\right)\).(d) Find \(P(33
Let \(X\) be a random variable having probability density function\[f(x)=2 x \text { for } \quad 0 \quad x \quad 1\](a) Find \(P\left(\begin{array}{ll}X \quad 75\end{array}\right)\).(b) Find
Let \(Z\) have the standard normal distribution.(a) Find \(P\left(\begin{array}{lll}0 & Z & 65\end{array}\right)\).(b) Find \(P\left(\begin{array}{ll}Z & 54\end{array}\right)\).(c) Find
Let \(Z\) have the standard normal distribution.(a) Find \(P\left(\begin{array}{lll}0 & Z & 152\end{array}\right)\).(b) Find \(P\left(\begin{array}{ll}Z & 211\end{array}\right)\).(c) Find
Let \(Y\) be normally distributed with mean \(=120\) and variance \({ }^{2}=64\).(a) Find \(P\left(\begin{array}{ll}Y & 130\end{array}\right)\).(b) Find \(P\left(\begin{array}{ll}Y &
Let \(Y\) be normally distributed with mean \(\quad=860\) and variance \({ }^{2}=576\).(a) Find \(P(Y \quad 900)\).(b) Find \(P\left(\begin{array}{ll}Y & 825\end{array}\right)\).(c) Find \(P(840
Let \(Y\) be distributed according to the beta(10 12) distribution.(a) Find \(\mathrm{E}[Y]\).(b) Find \(\operatorname{Var}[Y]\).(c) Find \(P(Y>5)\) using the normal approximation.
Let \(Y\) be distributed according to the beta \((1510)\) distribution.(a) Find \(\mathrm{E}[Y]\).(b) Find \(\operatorname{Var}[Y]\).(c) Find \(P(Y
Let \(Y\) be distributed according to the gamma(12 4) distribution.(a) Find \(\mathrm{E}[Y]\).(b) Find \(\operatorname{Var}[Y]\).(c) Find \(P(Y\)
Let \(Y\) be distributed according to the \(\operatorname{gamma}(265)\) distribution.(a) Find \(\mathrm{E}[Y]\).(b) Find \(\operatorname{Var}[Y]\).(c) Find \(P(Y>5)\)
In order to determine how e ective a magazine is at reaching its target audience, a market research company selects a random sample of people from the target audience and interviews them. Out of the
A city is considering building a new museum. The local paper wishes to determine the level of support for this project, and is going to conduct a poll of city residents. Out of the sample of 120
Sophie, the editor of the student newspaper, is going to conduct a survey of students to determine the level of support for the current president of the students' association. She needs to determine
You are going to take a random sample of voters in a city in order to estimate the proportion who support stopping the uoridation of the municipal water supply. Before you analyze the data, you need
In a research program on human health risk from recreational contact with water contaminated with pathogenic microbiological material, the National Institute of Water and Atmospheric Research (NIWA)
The same study found that \(y=12\) out of \(n=145\) samples identi ed as having a heavy environmental impact from dairy farms contained Giardia cysts.(a) What is the distribution of \(y\), the number
The same study found that \(y=10\) out of \(n=174\) samples identified as having a heavy environmental impact from pastoral (sheep) farms contained Giardia cysts.(a) What is the distribution of
The same study found that \(y=6\) out of \(n=87\) samples within municipal catchments contained Giardia cysts.(a) What is the distribution of \(y\), the number of samples containing \(G i\) ardia
Graph both posteriors on the same graph. What do you notice? What do you notice about the two posterior means and standard deviations? What do you notice about the two credible intervals for ?
Repeat the previous question with a uniform prior for .Data From previous Question We will use the Minitab macro BinoGCP or the R function binogcp to nd the posterior distribution of the
Graph the two posterior distributions on the same graph. What do you notice? What do you notice about the two posterior means and standard deviations? What do you notice about the two credible
Let be the proportion of students at a university who approve the government's policy on students' allowances. The students' newspaper is going to take a random sample of \(n=30\) students at a
The standard method of screening for a disease fails to detect the presence of the disease in \(15 \%\) of the patients who actually do have the disease. A new method of screening for the presence of
In the study of water quality in New Zealand streams, documented in McBride et al. (2002), a high level of Campylobacter was de ned as a level greater than 100 per \(100 \mathrm{ml}\) of stream
In the same study of water quality, \(n=145\) samples were taken from streams having a high environmental impact from dairying. Out of these \(y=9\) had a high Campylobacter level. Let be the true
In the same study of water quality, \(n=176\) samples were taken from streams having a high environmental impact from sheep farming. Out of these \(y=24\) had a high Campylobacter level. Let be the
In the same study of water quality, \(n=87\) samples were taken from streams in municipal catchments. Out of these \(y=8\) had a high Campylobacter level. Let be the true probability that a sample of
In Chapter 1 we learned that the frequentist procedure for evaluating a statistical procedure, namely looking at how it performs in the long-run, for a (range of) xed but unknown parameter values can
The number of particles emitted by a radioactive source during a ten second interval has the Poisson( ) distribution. The radioactive source is observed over ve non-overlapping intervals of ten
The number of claims received by an insurance company during a week follows a Poisson ( ) distribution. The weekly number of claims observed over a ten week period are: \(5,8,4,6,11,6,6,5,6,4\).(a)
The Russian mathematician Ladislaus Bortkiewicz noted that the Poisson distribution would apply to low-frequency events in a large population, even when the probabilities for individuals in the
The number of defects per 10 meters of cloth produced by a weaving machine has the Poisson distribution with mean . You examine 100 meters of cloth produced by the machine and observe 71 defects.(a)
We will use the Minitab macro PoisGamP, or poisgamp function in R, to nd the posterior distribution of the Poisson probability when we have a random sample of observations from a Poisson( )
Suppose we start with a Je reys' prior for the Poisson parameter .\[g()=\quad \frac{1}{2}\](a) What gamma( \(r v)\) prior will give this form?(b) Find the posterior distribution using the macro
Suppose we start with a \(\operatorname{gamma}(62)\) prior for . Find the posterior distribution using the macro PoisGamP in Minitab or or the function poisgamp in \(\mathrm{R}\).(a) Find the
Suppose we take an additional ve observations from the Poisson( ). They are:(a) Use the posterior from Computer Exercise 10.3 as the prior for the new observations and nd the posterior distribution
Suppose we use the entire sample of ten Poisson( ) observations as a single sample. We will start with the original prior from Computer Exercise 10.3 .(a) Find the posterior given all ten
We will use the Minitab macro PoisGCP, or the \(\mathrm{R}\) function poisgcp, to nd the posterior when we have a random sample from a Poisson( ) distribution and general continuous prior. Suppose we
You are the statistician responsible for quality standards at a cheese factory. You want the probability that a randomly chosen block of cheese labelled \(1 \mathrm{~kg}\) is actually less than 1
The city health inspector wishes to determine the mean bacteria count per liter of water at a popular city beach. Assume the number of bacteria per liter of water is normal with mean and standard
The standard process for making a polymer has mean yield \(35 \%\). A chemical engineer has developed a modified process. He runs the process on 10 batches and measures the yield (in percent) for
An engineer takes a sample of 5 steel I beams from a batch, and measures the amount they sag under a standard load. The amounts in \(\mathrm{mm}\) are:It is known that the sag is normal( \({ }^{2}\)
New Zealand was the last major land mass to be settled by human beings. The Shag River Mouth in Otago (lower South Island), New Zealand, is one of the sites of early human inhabitation that New
The Houhora site in Northland (top of North Island) New Zealand is one of the sites of early human inhabitation that New Zealand archeologists have investigated, in trying to determine when the
Suppose another 6 random observations come later. They are:Use NormDP in Minitab, or normdp in \(\mathrm{R}\), to nd the posterior distribution, where we will use the posterior after the rst ten
Instead, combine all the observations together to give a random sample of size \(n=16\), and use NormDP in Minitab, or normdp in \(\mathrm{R}\), to nd the posterior distribution where we go back the
Instead of thinking of a random sample of size \(n=16\), let's think of the sample mean as a single observation from its distribution.(a) What is the distribution of \(y\) ? Calculate the observed
Get to know your calculation tools and tables, to familiarize yourself with computing \(\phi_{(\mu, \sigma)}(x), \Phi_{(\mu, \sigma)}(x)\), and \(z_{\alpha}\). Write down your findings in a note book
Cumulative Normal distribution \(\Phi_{(\mu, \sigma)}\) and probability(a) \(X \sim \phi_{(0,1)}\); what is \(P(X \leq 1.43)\) ?(b) \(X \sim \phi_{(0,1)}\); what is \(P(X>1.43)\) ?(c) \(X \sim
Inverse cumulative Normal distribution \(z\)(a) Find \(z_{0.05}\).(b) Find \(z_{0.95}\).(c) Let \(X \sim \phi_{(2,1)}\). Find \(a\) such that \(P(X \leq a)=0.05\).(d) Let \(X \sim \phi_{(2,1)}\).
The Normal approximation(a) A discrete stochastic variable \(X\) has expected value \(\mu_{X}=3\) and \(\sigma_{X}=1.2\). Use the Normal approximation to find \(P(X \leq 4)\).(b) A continuous
Sums of Normally distributed stochastic variables(a) \(X_{1} \sim \phi_{(0,1)}, X_{2} \sim \phi_{(-1,3)}, X_{3} \sim \phi_{(2,4)}, X_{4} \sim \phi_{(2,2)}\). These four stochastic variables are
What is the probability that \(X
Let \(X\) be the stock price of SnowPeak Ltd, and let \(Y\) be the stock price of FjordWater Ltd. The prices of the two stocks are correlated, so \(Z=(X, Y)\) is binormally distributed with
You and Morty Matrix are campaigning on the high street for the coming election, handing out leaflets for the "Mathematics Party, because \(2+2=4\) ". You meet 1 sympathizer every 23 minutes. This
You are on probation with Statisticus Ltd for 4 weeks, as sales engineer. The waiting time between sales is exponentially distributed, but with parameter \(\lambda\) being a characteristic of the
You are measuring a radioactive material, and the number of minutes until the next click on the Geiger counter is exponentially distributed with parameter \(\lambda=0.25\).(a) What is the expected
At the bookstore where you are working, the number of minutes until the next sale of the book Home Engineering is exponentially distributed with parameter \(\lambda=\frac{1}{15}\).(a) What is
This Christmas, you have a seasonal job in a bookstore. \(T\), the number of minutes until the next customer asks about the book Home Engineering is exponentially distributed with parameter
You have studied Oh Mega's production of capacitors. Just as with the resistors in Example 10.3.3, their values follow a Normal distribution \(\phi_{(\mu, \sigma)}(x)\), and just as in the case of

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