Questions and Answers of Introduction To Statistical Investigations

Let $X_{1}, \ldots, X_{n}$ be a set of independent and identically distributed random variables from a distribution $F$ where \(E\left(\left|X_{1}ight|^{k}ight)
Let $X_{1}, \ldots, X_{n}$ be a set of independent and identically distributed random variables from a distribution $F$ with \(E\left(\left|X_{1}ight|^{2 k}ight)
Let $X_{1}, \ldots, X_{n}$ be a set of independent and identically distributed random variables from a distribution $F$ with \(E\left(\left|X_{1}ight|^{k}ight)
Consider an experiment that flips a fair coin 100 times. Define an indicator random variable $B_{n}$ so that\[B_{k}= \begin{cases}1 & \text { if the } k^{\text {th }} \text { flip is heads } \\ 0 &
Write a program in $\mathrm{R}$ that generates a sample $X_{1}, \ldots, X_{n}$ from a specified distribution $F$, computes the empirical distribution function of $X_{1}, \ldots, X_{n}$, and
Write a program in $\mathrm{R}$ that generates a sample $X_{1}, \ldots, X_{n}$ from a specified distribution $F$ and computes the sample mean $\bar{X}_{n}$. Use this program with
Write a program in $\mathrm{R}$ that generates independent $\operatorname{UnIFORM}(0,1)$ random variables $U_{1}, \ldots, U_{n}$. Define two sequences of random variables \(X_{1}, \ldots,
Write a program in $\mathrm{R}$ that generates a sample $X_{1}, \ldots, X_{n}$ from a specified distribution $F$, computes the empirical distribution function of $X_{1}, \ldots, X_{n}$,
Write a program in $\mathrm{R}$ that generates a sample from a population with distribution function\[F(x)= \begin{cases}0 & x
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables such that $X_{n}$ has a UNI$\operatorname{FORM}\left\{0, n^{-1}, 2 n^{-2}, \ldots, 1ight\}$ distribution for all \(n
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables where $X_{n}$ is an ExPo$\operatorname{NENTIAL}\left(\theta+n^{-1}ight)$ random variable for all $n \in \mathbb{N}$
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables such that for each $n \in \mathbb{N}$, $X_{n}$ has a $\operatorname{GAmma}\left(\alpha_{n}, \beta_{n}ight)$
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables where for each $n \in \mathbb{N}, X_{n}$ has an BERNoulli $\left[\frac{1}{2}+(n+2)^{-1}ight]$ distribution, and let
Let $\left\{X_{n}ight\}$ be a sequence of independent and identically distributed random variables where the distribution function of $X_{n}$ is\[F_{n}(x)= \begin{cases}1-x^{-\theta} & \text {
Suppose that $\left\{F_{n}ight\}_{n=1}^{\infty}$ is a sequence of distribution functions such that\[\lim _{n ightarrow \infty} F_{n}(x)=F(x)\]for all $x \in \mathbb{R}$ for some function
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables that converge in distribution to a random variable $X$ where $X_{n}$ has distribution function $F_{n}$ for all \(n
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables that converge in distribution to a random variable $X$ where $X_{n}$ has distribution function $F_{n}$ for all \(n
Let $g$ be a continuous and bounded function and let $\left\{F_{n}ight\}_{n=1}^{\infty}$ be a sequence of distribution functions such that $F_{n} \leadsto F$ as $n ightarrow \infty$ where
Consider the sequence of distribution functions $\left\{F_{n}ight\}_{n=1}^{\infty}$ where\[F_{n}(x)= \begin{cases}0 & x
Let $\left\{F_{n}ight\}_{n=1}^{\infty}$ be a sequence of distribution functions and let $F$ be a distribution function such that for each bounded and continuous function $g$,\[\lim _{n
Let $\left\{F_{n}ight\}_{n=1}^{\infty}$ be a sequence of distribution functions that converge in distribution to a distribution function $F$ as $n ightarrow \infty$. Prove that\[\lim _{n
In the context of the proof of Theorem 4.8 prove that\[\liminf _{n ightarrow \infty} F_{n}(x) \geq F(x-\varepsilon) .\] Theorem 4.8. Let {X} be a sequence of random variables that converge in
Prove the second result of Theorem 4.11. That is, let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables that converge weakly to a random variable $X$. Let
Prove the third result of Theorem 4.11. That is, let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables that converge weakly to a random variable $X$. Let
In the context of the proof of the first result of Theorem 4.11, prove that\[P\left(X_{n} \leq x-\varepsilon-cight) \leq G_{n}(x)+P\left(\left|Y_{n}-cight|>\varepsilonight) .\] Theorem 4.11
Use Theorem 4.11 to prove that if $\left\{X_{n}ight\}_{n=1}^{\infty}$ is a sequence of random variables that converge in probability to a random variable $X$ as $n ightarrow \infty$, then
Use Theorem 4.11 to prove that if $\left\{X_{n}ight\}_{n=1}^{\infty}$ is a sequence of random variables that converge in distribution to a real constant $c$ as $n ightarrow \infty$, then
In the context of the proof of Theorem 4.3, prove that\[\lim _{n ightarrow \infty}\left|\int_{a}^{b} g_{m}(x) d F(x)-\int_{a}^{b} g(x) d F(x)ight|for any $\delta_{\varepsilon}>0$. Theorem 4.3
Let $\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}$ be a sequence of $d$-dimensional random vectors where $\mathbf{X}_{n}$ has distribution function $F_{n}$ for all $n \in \mathbb{N}$ and let
Let $\mathbf{X}$ be a $d$-dimensional random vector with distribution function $F$. Let $g: \mathbb{R}^{d} ightarrow \mathbb{R}$ be a continuous function such that $|g(\mathbf{x})| \leq b$
Let $\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}$ be a sequence of $d$-dimensional random vectors that converge in distribution to a random vector $\mathbf{X}$ as $n ightarrow \infty$. Let
Prove the converse part of the proof of Theorem 4.17. That is, let $\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}$ be a sequence of $d$-dimensional random vectors and let $\mathbf{X}$ be a
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ and $\left\{Y_{n}ight\}_{n=1}^{\infty}$ be sequences of random variables where $X_{n}$ has a $\mathrm{N}\left(\mu_{n}, \sigma_{n}^{2}ight)$
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ and $\left\{Y_{n}ight\}_{n=1}^{\infty}$ be sequences of random variables that converge in distribution as $n ightarrow \infty$ to the random variables
In the context of the proof of Theorem 4.20, use Theorem A. 22 to prove that $\left[1-\frac{1}{2} n^{-1} t^{2}+o\left(n^{-1}ight)ight]^{n}=\left[1-\frac{1}{2} n^{-1} t^{2}ight]+o\left(n^{-1}ight)$
Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of independent and identically distributed random variables where $X_{n}$ has a $\operatorname{Exponential}(\theta)$ distribution. Prove
Prove that\[T^{-1} \frac{2}{9} \int_{-\infty}^{\infty} t^{2} \exp \left(-\frac{1}{4} t^{2}ight) d t=\frac{2}{3} \pi^{1 / 2} ho n^{-1 / 2},\]and\[T^{-1} \frac{1}{18} \int_{-\infty}^{\infty}|t|^{3}
Use Theorem 4.24 to prove Corollary 4.2. Theorem 4.24 (Berry and Esseen). Let {X}1 be a sequence of indepen- dent and identically distributed random variables from a distribution such that E(X) = 0
Prove Statement 3 of Theorem 4.26. Theorem 4.26. Let {X} be a sequence of independent random variables that have a common distribution F. Let p E (0, 1) and suppose that F is continuous at p. Then,
Prove Corollary 4.3. That is, let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of independent random variables that have a common distribution $F$. Let $p \in(0,1)$ and suppose that $F$
Prove Corollary 4.4. That is, let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of independent random variables that have a common distribution $F$. Let $p \in(0,1)$ and suppose that $F$
Write a program in $\mathrm{R}$ that simulates $b$ samples of size $n$ from an EXPONENTIAL(1) distribution. For each of the $b$ samples compute the minimum value of the sample. When the $b$
Write a program in $\mathrm{R}$ that simulates a sample of size $b$ from a Bino$\operatorname{MIAL}\left(n, n^{-1}ight)$ distribution and a sample of size $b$ from a POISson(1) distribution.
Write a program in $\mathrm{R}$ that generates a sample of size $b$ from a specified distribution $F_{n}$ (specified below) that weakly converges to a distribution $F$ as \(n ightarrow
Write a program in $\mathrm{R}$ that generates $b$ samples of size $n$ from a $\mathbf{N}\left(\mathbf{0}, \boldsymbol{\Sigma}_{n}ight)$ distribution
Write a program in $\mathrm{R}$ that generates $b$ samples of size $n$ from a specified distribution $F$. For each sample compute the statistic \(Z_{n}=n^{1 / 2}
Write a program in $\mathrm{R}$ that simulates $b$ samples of size $n$ from a distribution that has distribution function\[F(x)= \begin{cases}0 & xFor each sample, compute the sample
Wal-Mart buyers seek to purchase adequate supplies of various brands of toothpaste to meet the ongoing demands of their customers. In particular, Wal-Mart is interested in knowing the proportion of
Consider various characteristics of the U.S. civilian labor force provided in the file P3_60.XLS. In particular, examine the given unemployment rates taken across the United States.a. Characterize
Examine life expectations (in years) at birth for various countries across the world. These data can be found in the file P3_62.XLS. a. Generate an estimate of the typical human’s life span at
Tis problem focuses on the per capita circulation of daily newspapers in the United States during the period from 1991 to 1996. The file P3_63.XLS contains these data.a. Compare the yearly
Have the proportions of Americans receiving public aid changed in recent years? Explore this question through a careful examination of the data provided in the file P3_64.XLS. In particular, generate
The file P3_65.XLS contains the measured weight (in ounces) of a particular brand of ready-to-eat breakfast cereal placed in each of 500 randomly selected boxes by one of five different filling
The file P3_67.XLS contains the individual scores of students in two different accounting sections who took the same exam. Comment on the differences between exam scores in the two sections.
The file P3_68.XLS contains the monthly interest rates (from 1985 to 1995) on 3-month government T-bills. For example, in January 1985, 3-month T bills yielded 7.76% annual interest. To succeed in
Data on the numbers of insured commercial banks in the United States during the period 1990-1996 are given in the file P3_69.XLS. a. Compare these seven distributions of the numbers of U.S.
Educational attainment in the United States is the focus of this problem. Employ descriptive methods with the data provided in the file P3_70.XLS to characterize the educational achievements of
Continuing with the ShirtCo database found in the file P4_25.MDB, find all of the records from the Sales table that correspond to orders for over 500 items made by the customer Shirts R Us for the
Returning to the Fine Shirt Company, use the three tables contained in file P4_22.MDB to perform the following: a. Find all of the records from the Orders table that correspond to orders placed
ShirtCo would like to know the total amount spent by each of its customers on each of its products during each of the years from 1995 through 1998. Using the database given in the file P4_25.MDB,
ShirtCo would also like to know the proportions sold through each channel (i.e., wholesale versus retail) for each of its products during each quarter of the years from 1995 through 1998. Using the
The Fine Shirt Company would like to know what proportion of each customer’s total dollar purchases in 1999 came from buying Short-sleeve Seersucker shirts. Furthermore, it would like to compare
ShirtCo is trying to determine who was its biggest customer, as measured by total units sold, in 1998. Once ShirtCo determines which customer was responsible for the maximum level of total unit
A corporate executive officer is attempting to arrange a meeting of his three vice presidents for tomorrow morning. He believes that each of these three busy individuals, independently of the others,
Several students enrolled in a finance course subscribe to Money magazine. If two students are selected at random from this class, the probability that neither of the chosen students subscribes to
You have placed 30% of your money in investment A and 70% of your money in investment B. The annual returns on investments A and B depend on the state of the economy as shown in Table 5.35. Determine
There are three possible states of the economy during the next year (states 1, 2, and 3). The probability of each state of the economy, as well as the percentage annual return on IBM and Disney
There are three possible states of the economy during the next year (states 1, 2, and 3). The probability of each state of the economy, as well as the percentage annual return on P&G and US
The annual returns on stocks 1 and 2 for three possible states of the economy are given in Table 5.42. a. Find and interpret the correlation between stocks | and 2. b. Consider another stock
Wal-Mart buyers seek to purchase adequate supplies of various brands of toothpaste to meet the ongoing demands of its customers. In particular, Wal-Mart is interested in estimating the proportion of
Warren Dinner has invested in nine different investments. The returns on the different investments are probabilistically independent, and each return follows a normal distribution with mean $500 and
Consider a frame consisting of 500 households in a middle-class neighborhood that was the recent focus of an economic development study conducted by the local government. The data are in the file
The manager of a local supermarket wants to know the average amount (in dollars) customers spend at his store on Fridays. He would like to study the buying behavior of each customer who makes a
The manager of a local supermarket wants to know the average amount (in dollars) customers spend at his store on Fridays. He would like to study the buying behavior of each customer who makes a
Auditors of a particular bank are interested in comparing the reported value of customer savings account balances with their own findings regarding the actual value of such assets. Rather than
The manager of a local supermarket wants to estimate the average amount customers spend at his store on Fridays. He would like to study the buying behavior of each customer who makes a purchase at
The Hite Report was Sheri Hite’s survey of the attitudes of American women toward sexuality. She sent out over 100,000 surveys; each contained multiple-choice and open-ended questions. These
Repeat Problem 65, but now stratify the consumers in the given frame by age rather than by gender. How does this modification affect your answers to the questions posed in parts b and c? Finally,
Wal-Mart buyers seek to purchase adequate supplies of various brands of toothpaste to meet the ongoing demands of their customers. In particular, Wal-Mart is interested in knowing the proportion of
Suppose that you are an entrepreneur interested in establishing a new Internet-based sports information service. Furthermore, suppose that you have gathered basic demographic information on a large
Continuing the previous problem, suppose that SteelCo can pay money to reduce the standard deviation of the process. It costs e100D dollars to reduce the standard deviation from 0.025 to 0.025 — d.
Suppose that 60% of all people prefer Coke to Pepsi. We randomly choose 500 people and ask them if they prefer Coke to Pepsi. What is the probability that our survey will (erroneously) indicate that
Consider again SweetTooth Candy Company’s decision problem described in Problem 1. Use the PrecisionTree add-in to identify the strategy that minimizes SweetTooth’s expected cost of meeting its
Consider again Carlisle Tire and Rubber’s decision problem described in Problem 2. Use the PrecisionTree add-in to identify the strategy that maximizes this tire manufacturer’s expected profit.
Consider again the landowner’s decision problem described in Problem3. Use the Precision Tree add-in to identify the strategy that maximizes the landowner’s expected net earnings from this
The lifetime of a washing machine is normally distributed with mean 4 years. Only 15% of all washing machines last at least 5 years. What is the standard deviation of the lifetime of a washing
You have been told that the distribution of regular unleaded gasoline prices over all gas stations in Indiana is normally distributed with mean $1.25 and standard deviation $.075, and you have been
Suppose that the number of ounces of soda put into a Pepsi can is normally distributed with jt = 12.05 ounces and o = 0.03 ounce. a Legally, acan must contain at least 12 ounces of soda. What
Suppose that 52% of all registered voters prefer Bill Clinton to Bob Dole. (You may substitute the names of the current presidential candidates!) a Inarandom sample of 100 voters, what is the
A lightbulb manufacturer wants to estimate the number of defective bulbs contained in a typical box shipped by the company. Production personnel at this company have recorded the number of defective
The employee benefits manager of a small private university would like to estimate the proportion of full-time employees who prefer adopting the first (i.e., plan A) of three available health care
A manufacturing company’s quality control personnel have recorded the proportion of defective items for each of 500 monthly shipments of one of the computer components that the company produces.
A finance professor has just given a midterm examination in her corporate finance course. In particular, she is interested in learning how her large class of 100 students performed on this exam. The
Consider a frame consisting of 500 households in a middle-class neighborhood that was the recent focus of an economic development study conducted by the local government. The data are in the file
A real estate agent has received data on 150 houses that were recently sold in a suburban community. Included in this data set are observations for each of the following variables: the appraised
Consider the average time (in minutes) it takes citizens of each of 325 metropolitan areas across the United States to travel to work and back home each day. The data are in the file P2_11.XLS. Use
Find a 95% confidence interval for the mean total cost of all customer orders. Then do this separately for each of the four regions. Create side-by-side boxplots of total cost for the four regions.
Find a 95% confidence interval for the mean amount of all Rebco’s bills. Do the same for each customer size separately.
Find a 95% confidence interval for the mean number of days it takes Rebco’s customers (as a combined group) to pay their bills. Do the same for each customer size separately. Create a boxplot for
The following probability distribution represents the payout for a game. Suppose $2 is added to all the payouts. How would the mean and standard deviation change from those of the original

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