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statistics
elementary statistics a step by step approach

Questions and Answers of Elementary Statistics A Step By Step Approach

Show that the log-likelihood in equation (12.2) has a maximum at \(\widehat{\mu}=\bar{y}\). n L() Inf(y,) = (-+y; In - In y;!). i=1 i=1 (12.2)
For the data in Table 12.1, confirm that the Pearson statistic in equation (12.3) is 41.98 . Table 12.1(12.3) Count Observed (j) (nj) Fitted Counts Using the Poisson Distribution (np;) 01234 6,996
Consider a Poisson regression. Let \(e_{i}=y_{i}-\widehat{\mu}_{i}\) denote the \(i\) th ordinary residual. Assume that an intercept is used in the model so that one of the explanatory variables
a. Assume that \(y_{1}, \ldots, y_{n}\) are i.i.d. with a negative binomial distribution with parameters \(r\) and \(p\). Determine the maximum likelihood estimators.b. Use the sampling mechanism in
Verify the entries in Table 13.8 for the gamma distribution. Specifically:a. Show that the gamma is a member of the linear exponential family of distributions.b. Describe the components of the linear
Consider a random walk \(\left\{y_{t}\right\}\) as the partial sum of a white noise process \(\left\{c_{t}\right\}\).a. Show that the \(l\)-step forecast error is
A mutual fund has provided investment yield rates for five consecutive years as follows:Determine \(r_{1}\) and \(r_{2}\), the lag 1 and lag 2 autocorrelation coefficients.Determine \(r_{1}\) and
The Durbin-Watson statistic is designed to detect autocorrelation and is defined by\[D W=\frac{\sum_{t=2}^{T}\left(y_{t}-y_{t-1}\right)^{2}}{\sum_{t=1}^{T}\left(y_{t}-\bar{y}\right)^{2}} .\]a. Derive
Consider the Chapter 2 linear regression model formulas with \(y_{t-1}\) in place of \(x_{t}\), for \(t=2, \ldots, T\).a. Provide an exact expression for \(b_{1}\).b. Provide an exact expression for
Begin with the \(A R(1)\) model as in equation (8.1).a. Take variances of each side of equation (8.1) to show that \(\sigma_{y}^{2}\left(1-\beta_{1}^{2}\right)=\) \(\sigma^{2}\), where
Consider forecasting with the \(A R(1)\) model.a. Use the forecasting chain rule in equation (8.4) to show\[y_{T+k}-\widehat{y}_{T+k} \approx \varepsilon_{T+k}+\beta_{1}
These data consist of the 503 daily returns for the calendar years 2005 and 2006 of the S\&P value-weighted index. (The data file contains additional years - this exercise uses only 2005 and 2006
Consider an underlying linear model, \(y_{i}^{*}=\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}+\epsilon_{i}^{*}\), where \(\epsilon_{i}^{*}\) is normally distributed with mean zero and variance
Under the random utility interpretation, an individual with utility \(U_{i j}=\) \(u_{i}\left(V_{i j}+\epsilon_{i j}\right)\), where \(j\) may be 1 or 2 , selects category corresponding to \(j=1\)
a. Begin with one population and assume that \(y_{1}, \ldots, y_{n}\) is an i.i.d. sample from a Bernoulli distribution with mean \(\pi\). Show that the maximum likelihood estimator of \(\pi\) is
Fitted Values. Let \(\left.\widehat{y}_{i}=\pi\left(\mathbf{x}_{i}^{\prime} \mathbf{b}_{M L E}\right)\right)\) denote the \(i\) th fitted value for the logit function. Assume that an intercept is
Beginning with the score equations (11.4), verify the expression for the logit case in equation (11.5). a L() = x; (y; (x;)) i=1 '(x(B) (x)(1-(x)) = 0, (11.4)
a. Beginning with the score function for the logit case in equation (11.5), show that the information matrix can be expressed as\[\mathbf{I}(\boldsymbol{\beta})=\sum_{i=1}^{n} \sigma_{i}^{2}
Automobile Injury Insurance Claims. Refer to the description in Exercise 1.5 .We consider \(n=1,340\) bodily injury liability claims from a single state using a 2002 survey conducted by the Insurance
The race track is a fascinating example of financial market dynamics at work. Let's go to the track and make a wager. Suppose that, from a field of 10 horses, we simply want to pick a winner. In the
We continue our study of term life insurance demand from Chapters 3 and 4. Specifically, we examine the 2004 Survey of Consumer Finances (SCF), a nationally representative sample that contains
Much like the medical and legal fields, members of the actuarial profession face interesting problems and are generally well compensated for their efforts in resolving these problems. Also like the
Consider the following "case-control" sample selection method for binary dependent variables. Intuitively, if we are working with a problem in which the event of interest is rare, we want to make
In addition to the size variables, we also have information on several binary variables. The variable URBAN is used to indicate the facility's location. It is one if the facility is located in an
a. Run a regression of LNPAID on AGE. Is AGE a statistically significant variable? To respond to this question, use a formal test of hypothesis. State your null and alternative hypotheses,
a. Are the quadratic terms important? Consider a linear model of LNEXPENSES on 12 explanatory variables. For the explanatory variables, include assets, GROUP, both versions of losses and gross
a. Consider the regression using three explanatory variables, FERTILITY, PUBLICEDUCATION, and LNHEALTH that you did in Exercise 3.3.6. Test whether PUBLICEDUCATION and LNHEALTH are jointly
You are doing regression with one explanatory variable and so consider the basic linear regression model \(y_{i}=\beta_{0}+\beta_{1} x_{i}+\varepsilon_{i}\).a. Show that the \(i\) th leverage can be
Consider the output of a regression using one explanatory variable on \(n=3\) observations. The residuals and leverages are:Compute the PRESS statistic. i 1 2 3 Residuals ei Leverages hi 3.181 -6.362
a. Begin with the data from \(n=185\) countries throughout the world that have valid (nonmissing) life expectancies. Plot the life expectancy versus the gross domestic product and private
Specifically, we examine the 2004 Survey of Consumer Finances (SCF), a nationally representative sample that contains extensive information on assets, liabilities, income, and demographic
Do insurance companies use race as a determining factor when making insurance available? Fienberg (1985) gathered data from a report issued by the U.S. Commission on Civil Rights about the number of
The University of Wisconsin at Madison completed a study titled “GenderEquity Study of Faculty Pay,”dated June 5, 1992. The main purpose of the study was to determine whether women are treated
Consider the following datasetFit a regression line using the method of least squares. Determine \(r, b_{1}\), and \(b_{0}\). i 1 xi 23 2 Yi 26 -6 3 7. 91 4 6
Use the following steps to show that \(r\) is bounded by -1 and 1 (These steps are due to Koch, 1990).a. Let \(a\) and \(c\) be generic constants. Verify \[\begin{aligned}0 & \leq \frac{1}{n-1}
Show that the intercept term, \(b_{0}\), can be expressed as a weighted sum of the dependent variables. That is, show that \(b_{0}=\sum_{i=1}^{n} w_{i, 0} y_{i}\). Further, express the weights in
a. Using algebra, establish an alternative expression\[b_{1}=\frac{\sum_{i=1}^{n} \text { weight }_{i} \times \text { slope }_{i}}{\sum_{i=1}^{n} \text { weight }_{i}} .\]Here, slope \(_{i}\) is the
Consider the model \(y_{i}=\beta_{1} x_{i}+\varepsilon_{i}\), that is, regression with one explanatory variable without the intercept term. This model is called regression through the origin because
Suppose that \(x_{i}\) only takes on the values 0 and 1. Out of the \(n\) observations, \(n_{1}\) take on the value \(x=0\). The \(n_{1}\) observations have an average \(y\) value of \(\bar{y}_{1}\).
This exercise considers nursing home data provided by the Wisconsin Department of Health and Family Services (DHFS).Part 1: Use cost-report year 2000 data, and do the following analysis.a.
Suppose that, for a sample size of \(n=3\), you have \(e_{2}=24\) and \(e_{3}=-1\). Determine \(e_{1}\).
Suppose that \(r=0, n=15\), and \(s_{y}=10\). Determine \(s\).
Use the following steps to establish a relationship between the coefficient of determination and the correlation coefficient.a. Show that
Show that the average residual is zero, that is, show that \(n^{-1} \sum_{i=1}^{n} e_{i}=0\).
Correlation between Residuals and Explanatory Variables. Consider a generic sequence of pairs of numbers \(\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)\) with the correlation
Correlation and \(t\)-Statistics. Use the following steps to establish a relationship between the correlation coefficient and the \(t\)-statistic for the slope.a. Use algebra to check that
Effects of an Unusual Point. You are analyzing a data set of size \(n=100\). You have just performed a regression analysis using one predictor variable and notice that the residual for the 10th
Consider a dataset consisting of 20 observations with the following summary statistics: \(\bar{x}=0, \bar{y}=9, s_{x}=1\), and \(s_{y}=10\). You run a regression using using one variable and
The data in Table 2.9 is due to Anscombe (1973). The purpose of this exercise is to demonstrate how plotting data can reveal important information that is not evident in numerical summary
This exercise considers nursing home data provided by the Wisconsin Department of Health and Family Services (DHFS) and described in Exercises 1.2 and 2.10. You decide to examine the relationship
As a financial analyst, you want to convince a client of the merits of investing in firms that have just entered a stock exchange, as an initial public offering (IPO). Thus, you gather data on 116
We continue the analysis begun in Exercise 1.7 by examining the relation between y = LIFEEXP and x = FERTILITY, shown in Figure 2.12. Fit a linear regression model of LIFEEXP using the explanatory
Consider a fictitious dataset of \(n=100\) observations with \(s_{y}=80\). We run a regression with three explanatory variables to get \(s=50\).a. Calculate the adjusted coefficient of determination,
Section2.1 described a sample of \(n=50\) geographic areas (Zip codes) containing sales data on the Wisconsin state lottery ( \(y=\) SALES). In that section, sales were analyzed using a basic linear
National Life Expectancies. We continue the analysis begun in Exercises 1.7and 2.22Now fit a regression model on LIFEEXP using three explananatural logarithmic transform of PRIVATEHEALTH).a.
This exercise considers data from the Medical Expenditure Panel Survey (MEPS), conducted by the U.S. Agency of Health Research and Quality. MEPS is a probability survey that provides nationally
This exercise considers nursing home data provided by theWisconsin Department of Health and Family Services (DHFS).The State of Wisconsin Medicaid program funds nursing home care for individuals
We consider automobile injury claims data using data from the Insurance Research Council (IRC), a division of the American Institute for Chartered Property Casualty Underwriters and the Insurance
This exercise considers data from the Medical Expenditure Panel Survey (MEPS) described in Exercise 1.1 and Section 11.4. Our dependent variable consists of the number of outpatient (COUNTOP) visits.
This exercise considers the data described in the Section 13.2.2 ratemaking classification example using data in Table 13.3Table 13.3 .a. Fit a gamma regression model using a log-link function with
Verify that the Tweedie distribution is a member of the linear exponential family of distributions by checking equation (13.9). In particular, provide an expression for \(S(y, \phi)\) (note that
Assume that \(y\) is normally distributed with mean \(\mu\) and variance \(\sigma^{2}\). Let \(\phi(\cdot)\) and \(\Phi(\cdot)\) be the standard normal density and distribution functions,
Use equation (17.2) to establish the following distributional relationships that are helpful for calculating quantiles.a. Assume that \(y_{0}=\alpha_{1} F / \alpha_{2}\), where \(F\) has an
Consider a GB2 probability density function given in equation (17.3).a. Reparameterize the distribution by defining the new parameter \(\theta=e^{\mu}\). Show that the density can be expressed
Recall that the density of a gamma distribution with shape parameter \(\alpha\) and scale parameter \(\theta\) has a density given by \(\mathrm{f}(y)=\) \(\left[\theta^{\alpha}
The data in Table 19.5 originate from the 1991 edition of the Historical Loss Development Study, published by the Reinsurance Association of American. These data have been widely used to illustrate
The data in Table 19.8 are from Gamage et al. (2007). These data are for 36 months of medical-care payments, from January 2001 through December 2003, inclusive. These are payments for medical-care
Determinants of CEO Compensation. Chief executive officer (CEO) compensation varies significantly from firm to firm. For this exercise, you will report on a sample of firms from a survey by Forbes
Two Population Poissons. We can express the two population problem in a regression context using one explanatory variable. Specifically, suppose that \(x_{i}\) only takes on the values of zero and
Verify the log-likelihood in equation (16.4) for the Tobit model. In L = = In { 1-0 (x-di)} 1:y=di 122. + (y; - x) 02 (16.4) i:y;>di
Verify the likelihood in equation (16.5) for the two-part model. n2. (16.5) -(-)-(-2)/02 L = [] {(p;)" (1 p; )'-'} [[ ( i=1 ri=1
Derive the likelihood for the tobit type II model. Show that your likelihood reduces to equation (16.5) in the case of uncorrelated disturbance terms. n2. (16.5) -(-)-(-2)/02 L = [] {(p;)" (1 p;
A Perfect Relationship, yet Zero Correlation. Consider the quadratic relationship \(y=x^{2}\), with dataa. Produce a rough graph for this dataset.b. Check that the correlation coefficient is \(r=0\).
Consider two variables, \(y\) and \(x\). Do a regression of \(y\) on \(x\) to get a slope coefficient that we call \(b_{1, x, y}\). Do another regression of \(x\) on \(y\) to get a slope coefficient
a. Show that\[s_{y}^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}=\frac{1}{n-1}\left(\sum_{i=1}^{n} y_{i}^{2}-n \bar{y}^{2}\right) .\]b. Follow the same steps to show
. Consider a fictitious dataset of \(n=100\) observations with \(s_{y}=80\). We run a regression with three explanatory variables to get \(s=50\). We also geta. Determine the standard error of
As an actuarial analyst, you are working with a large insurance company to help it understand claims distribution for private passenger automobile policies. You have available claims data for a
Suppose that you are an employee benefits actuary working with a medium-sized company in Wisconsin. This company is considering offering, for the first time in its industry, hospital insurance
Like other businesses, insurance companies seek to minimize expenses associated with doing business to enhance profitability. To study expenses, this exercise examines a random sample of 500
Who is doing health care right? Health-care decisions are made at the individual, corporate, and government levels. Virtually every person, corporation, and government has a different perspective on
On what distribution does one-way ANOVA rely?
For what is one-way ANOVA used?
A CNN/USA TODAY poll conducted by Gallup asked a sample of employed Americans the following question: “Which do you enjoy more, the hours when you are on your job, or the hours when you are not on
A CNN/USA TODAY poll conducted by Gallup asked a sample of employed Americans the following question: “Which do you enjoy more, the hours when you are on your job, or the hours when you are not on
A CNN/USA TODAY poll conducted by Gallup asked a sample of employed Americans the following question: “Which do you enjoy more, the hours when you are on your job, or the hours when you are not on
A CNN/USA TODAY poll conducted by Gallup asked a sample of employed Americans the following question: “Which do you enjoy more, the hours when you are on your job, or the hours when you are not on
We have given the number of possible values for two variables of a population. For each exercise, determine the maximum number of expected frequencies that can be less than 5 in order that Assumption
We have given the number of possible values for two variables of a population. For each exercise, determine the maximum number of expected frequencies that can be less than 5 in order that Assumption
We have given the number of possible values for two variables of a population. For each exercise, determine the maximum number of expected frequencies that can be less than 5 in order that Assumption
We have given the number of possible values for two variables of a population. For each exercise, determine the maximum number of expected frequencies that can be less than 5 in order that Assumption
We have given the number of possible values for two variables of a population. For each exercise, determine the maximum number of expected frequencies that can be less than 5 in order that Assumption
The U.S. Census Bureau compiles information on the money income of people by type of residence and publishes its finding in Current Population Reports. Independent simple random samples of people
There is convincing evidence that breastmilk containing antioxidants is important in the prevention of diseases in infants. Researchers A. Xavier et al. studied the effects of storing breastmilk on
In Exercise, we have provided a sample mean, sample standard deviation, and sample size. In each case, use the one-mean t-test to perform the required hypothesis test at the 5% significance level.x̄
The U.S. Department of Justice, Office of Justice Programs, and Bureau of Justice Statistics provides information on prison sentences in the document National Corrections Reporting Program. A random
With the advent of high-speed computing, new procedures have been developed that permit statistical inferences to be performed under less restrictive conditions than those of classical procedures.
In Example 7.9, we conducted a simulation to check the plausibility of the central limit theorem. The variable under consideration there is household size, and the population consists of all U.S.
Use the technology of your choice toa. Decide whether use of the linear correlation coefficient as a descriptive measure for the data is appropriate. If so, then also do parts (b) and (c).b. Obtain
Use the technology of your choice to perform the following tasks.a. Decide whether finding a regression line for the data is reasonable. If so, then also do parts (b)–(d).b. Obtain the coefficient
Following are the data on age and crown-rump length for fetuses from Exercises 4.62 and 4.102.Exercise 4.62Crown-Rump Length. In the article “The Human Vomeronasal Organ. Part II: Prenatal

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