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business
statistics for business and economics

Questions and Answers of Statistics For Business And Economics

The following data were used to construct the histograms of the number of days required to fill orders for Dawson Supply, Inc., and J.C. Clark Distributors (see Figure 3.2).Dawson Supply Days for
How do grocery costs compare across the country? Using a market basket of 10 items including meat, milk, bread, eggs, coffee, potatoes, cereal, and orange juice, Where to Retire magazine calculated
Scores turned in by an amateur golfer at the Bonita Fairways Golf Course in Bonita Springs, Florida, during 2005 and 2006 are as follows:2005 Season: 74 78 79 77 75 73 75 77 2006 Season: 71 70 75 77
The following times were recorded by the quarter-mile and mile runners of a university track team (times are in minutes).Quarter-Mile Times: .92 .98 1.04 .90 .99 Mile Times: 4.52 4.35 4.60 4.70 4.50
Suppose the data have a bell-shaped distribution with a mean of 30 and a standard deviation of 5.Use the empirical rule to determine the percentage of data within each of the following ranges:a. 20
The national average for the math portion of the College Board’s Scholastic Aptitude Test(SAT) is 515 (The World Almanac, 2009). The College Board periodically rescales the test scores such that
A sample of 10 NCAA college basketball game scores provided the following data (USA Today, January 26, 2004).Winning Winning Team Points Losing Team Points Margin Arizona 90 Oregon 66 24 Duke 85
Consumer Reports posts reviews and ratings of a variety of products on its website. The following is a sample of 20 speaker systems and their ratings. The ratings are on a scale of 1 to 5, with 5
Show the box plot for the data in exercise 36.
A data set has a first quartile of 42 and a third quartile of 50.Compute the lower and upper limits for the corresponding box plot. Should a data value of 65 be considered an outlier?Applications
Annual sales, in millions of dollars, for 21 pharmaceutical companies follow.8408 1374 1872 8879 2459 11413 608 14138 6452 1850 2818 1356 10498 7478 4019 4341 739 2127 3653 5794 8305a. Provide a
The Philadelphia Phillies defeated the Tampa Bay Rays 4 to 3 to win the 2008 major league baseball World Series (The Philadelphia Inquirer, October 29, 2008). Earlier in the major league baseball
Five observations taken for two variables follow.xi 4 6 11 3 16 yi 50 50 40 60 30a. Develop a scatter diagram with x on the horizontal axis.b. What does the scatter diagram developed in part (a)
Five observations taken for two variables follow.xi 6 11 15 21 27 yi 6 9 6 17 12a. Develop a scatter diagram for these data.b. What does the scatter diagram indicate about a relationship between x
Nielsen Media Research provides two measures of the television viewing audience: a television program rating, which is the percentage of households with televisions watching a program, and a
A department of transportation’s study on driving speed and miles per gallon for midsize automobiles resulted in the following data:Speed (Miles per Hour) 30 50 40 55 30 25 60 25 50 55 Miles per
Consider the sample data in the following frequency distribution.a. Compute the sample mean.b. Compute the sample variance and sample standard deviation.Applications
According to an annual consumer spending survey, the average monthly Bank of America Visa credit card charge was $1838 (U.S. Airways Attaché Magazine, December 2003). A sample of monthly credit card
The U.S. Census Bureau provides statistics on family life in the United States, including the age at the time of first marriage, current marital status, and size of household(U.S. Census Bureau
Dividend yield is the annual dividend per share a company pays divided by the current market price per share expressed as a percentage. A sample of 10 large companies provided the following dividend
The U.S. Department of Education reports that about 50% of all college students use a student loan to help cover college expenses (National Center for Educational Studies, January 2006). A sample of
Small business owners often look to payroll service companies to handle their employee payroll. Reasons are that small business owners face complicated tax regulations and penalties for employment
Public transportation and the automobile are two methods an employee can use to get to work each day. Samples of times recorded for each method are shown. Times are in minutes.Public Transportation:
The National Association of Realtors reported the median home price in the United States and the increase in median home price over a five-year period (The Wall Street Journal,January 16, 2006). Use
Travel + Leisure magazine presented its annual list of the 500 best hotels in the world(Travel + Leisure, January 2009). The magazine provides a rating for each hotel along with a brief description
Consider a $4^{2} \times 3^{2} \times 2$ factorial design.a. How many factors are included in this design?b. How many levels are included in each factor?c. How many experimental conditions, or runs,
Consider a $2^{2}$ factorial experiment with factors A and B. Show that $I N T(A, B)=I N T(B, A)$. That is, the interaction is symmetric in $\mathrm{B}$ and A.
Consider a $2^{3}$ experiment with factors A, B, and C. Show that\[\begin{aligned}I N T(A, B, C) & =\frac{1}{2}[I N T(A, C \mid B+)-I N T(A, C \mid B-)] \\& =\frac{1}{2}[I N T(C, B \mid A+)-I N T(C,
Example 6.1 usedsum(wtlossdat ["B"] *wtlossdat["y"])/4to compute the main effect of $\mathrm{C}$.a. Explain why the espression is divided by 4 ?b. Write an $\mathrm{R}$ function to calculate the main
Show that the variance of the ith run in a 23 design with two replications iswhere \(y_{i j}\) is \(j^{\text {th }}\) observation in \(i^{\text {th }}\) experimental run. (yil - Yiz) 2
Show that the variance of a factorial effect in a 2k design with m replications is4σ2Nwhere \(N=m \cdot 2^{k}\) and \(\sigma^{2}\) is the variance of each outcome. 40 N
Assume that $y_{i j}$ are i.i.d and follow a normal distribution with variance $\sigma^{2}$. Under the null hypothesis that a factorial effect 0 ,a. Show that\[\frac{\bar{y}_{+}-\bar{y}_{-}}{s / 2}
Use $\mathrm{R}$ to calculate the three-way interaction effects from the experiment in Example 6.2 . Interpret these interaction terms.Data from Example 6.2 Example 6.2 (Silk Study). Silk fibroin is
This exercise is based on Section 5.2 of Box et al. [2005]. An experiment for optimizing response yield (y) in a chemical plant operation was conducted. The three factors considered are temperature
Interpret the intercept term in the linear regression model represented in Example 6.1 .Data from Example 6.1 Example 6.1. An investigator is interested in examining three components of a weight loss
Suppose that a $2^{2}$ factorial design studying factors A and B was conducted. An investigator fits the model\[y_{i}=\beta_{0} x_{i 0}+\beta_{1} x_{i 1}+\beta_{2} x_{i 2}+\beta_{3} x_{i
Suppose that linear regression is used to estimate factorial effects for a $2^{k}$ design by doubling the estimated regression coefficients. a. When is it possible to estimate the standard error of
Suppose $Y$ has a half-normal distribution with variance $\sigma^{2}=1$. Let $y_{p}$ be the $p^{t h}$ quantile of $Y$ such that $p=F\left(y_{p}\right)$ and $\Phi$ be the CDF of the standard normal
In Section 6.5 .2 Computation Lab: Normal Plots in Unreplicated Factorial Designs, the factorial effects were sorted based on their magnitudes before plotting the normal quantile plot shown in Figure
Reconstruct the half-normal plot shown in Figure 6.8 using geom_qq() and geom_qq_line().Data from Figure 6.8 Factorial Effects B:C -2 B -1.5 -1.0 -0.5 0.0 A:B AB.C 2.0- A A:C B Absolute Factorial
The planning matrix from a $2^{k-p}$ design is below. The four factors investigated are A, B, C, and D.TABLE 6.13 : Planning Matrix for Exercise 6.16b. What is the defining relation? What is the
Use the $\mathrm{FrF} 2():: \mathrm{FrF} 2()$ function to generate the design matrix in the standard order for a $2^{3}$ design with blocks generated using the three-way interaction.
An investigator is considering two blocking schemes for a $2^{4}$ design with 4 blocks. The two schemes are listed below.Scheme 1: $B_{1}=134, B_{2}=234$• Scheme 2: $B_{1}=12, B_{2}=13,
This exercise is based on Section 5.1 of Wu and Hamada [2011]. An experiment to improve a heat treatment process on truck leaf springs was conducted. The heat treatment that forms the camber in leaf
Suppose $N$ experimental units are randomly assigned to treatment or control by tossing a coin. A unit is assigned to treatment if the coin toss comes up heads. Assume that the probability of tossing
Consider the two treatment assignment mechanisms (7.1) and (7.2). What is the probability \(P\left(T_{i}=1\right)\) in each case? What is \(P\left(T_{i}=1 \mid Y_{i}(0)>\right.\)
Suppose you are asked to design a study to evaluate the effect of the presence of vending machines in schools on childhood obesity. Describe randomized and non-randomized studies to evaluate this
The table below describes a hypothetical experiment on 2400 persons. Each row of the table specifies a category of person, as defined by their pre-treatment predictor $x$, treatment indicator $T$,
What is the definition of ignorable treatment assignment?a. Give an example of a study where the treatment assignment is ignorable.b. Give an example of a study where the treatment assignment is
An observational study to evaluate the effectiveness of supplementing a reading program with a television show was conducted in several schools in grade 4. Some classroom teachers chose to supplement
The following questions refer to Example 7.9 and Section 7.7.8. a. Evaluate the parallelism assumption in mod_ancova. What do you conclude?b. Compute the adjusted difference in mean weight gain using
Consider the data from Example 7.9. In Section 7.7.4 Computation Lab: Nearest Neighbour Propensity Score Matching,marital1982 was included when fitting the propensity score model propmod_nhefs.
Figure 2.3 displays the frequency distribution from Example 2.5. Create a similar plot using ggplot () to display the relative frequency distribution for each treatment group instead of the frequency
Reproduce Figure 2.4 using geom_histogram(aes(y = ..density..)) as shown below for Example 2.5. ggplot_build() extracts the computed values for the histogram. Use the extracted vales to confirm that
In Section 2.3, 20 patients for Example 2.5 were randomly selected from a population of 25,000 .a. Explain why the probability of choosing a random sample of 20 from 25,000
The treatment assignment displayed in Table 2.5 is from one treatment assignment out of $\left(\begin{array}{l}20 \\ 10\end{array}\right)$ possible random treatment assignments. a. $\mathrm{R}$ can
In this exercise you will use $\mathrm{R}$ to compute the normal density and the cumulative distribution functions using dnorm and pnorm.a. For any given values of mu, sigma, and y, what value does
Suppose $X \sim N(0,1)$ and $W_{n} \sim \chi_{n}^{2}$ independently for any positive integer $n$. Let $V_{n}=X / \sqrt{W_{n}} / n$.a. We know $V_{n} \sim t_{n}$. Show that $V_{n}^{2}$ follows an
Explain why the reference line in Figure 2.12 uses the first and third quartiles from the Unif $[0,1]$, and ordered sample.Data from Figure 2.12 2+ X (Sample Quantiles) N GO Unif[0,1] sample N(0,1)
Recall that simulated squared values from Exercise 2.7 follow F distributions. Use geom_qq and geom_qq_line to plot Q-Q plots of the simulated squared values against the appropriate $\mathrm{F}$
We generated Figure 2.16 by first computing the mean values for each column and then pivoting the values into a single column of mean values. Replicate the data generation steps and use the
A chemist has seven light objects to weigh on a balance pan scale. The standard deviation of each weighing is denoted by $\sigma$. In a 1935 paper, Frank Yates [Yates, 1935] suggested an improved
Does life satisfaction change by region over time? Use the lifesat_childmort data from Example 2.15 to explore this question.Data from Example 2.15The World Happiness Report [Helliwell et al., 2019]
Suppose $X_{1} \sim N(10,25)$ and $X_{2} \sim N(5,4)$ in a population. You randomly select 100 samples from the population and assign treatment $A$ to half of the sample and $B$ to the rest. Simulate
Identify treatments and experimental units in the following scenarios.a. City A would like to evaluate whether a new employment training program for the unemployed is more effective compared to the
A study has three experimental units and two treatments-A and B. List all possible treatment assignments for the study. How many are there? In general, show that there are $2^{N}$ possible treatment
Consider the scenario in Example 3.1, and suppose that an investigator only has enough fertilizer A to use on four plots. Answer the following questions. a. What is the probability that an
Show that the two-sided p-value is \(1-\hat{F}_{T}\left(\left|t^{*}\right|\right)+\hat{F}_{T}\left(-\left|t^{*}\right|\right)\), where \(\hat{F}_{T}\) is the ECDF of the randomization distribution of
The actual confidence level conf_level does not equal the theoretical confidence level 0.01 in Example 3.2. Explain why.Data from Example 3.2Data form Figure 3.3 Example 3.2 (Confidence interval for
Consider Example 3.5. For each of the 10 boys, we randomly assigned the left or right sole to material A and the remaining side to B. Use $R$ 's sample function to simulate a treatment
Recall that the randomization test for the data in Example 3.5 fails to find evidence of a significant increase in the amount of wear with material B. Does this mean that material B has equivalent
Consider the study from Example 3.4. Recall that the clinical trial consists of 450 patients. 150 of the patients have stage I cancer and the rest have stages II-IV cancer. In Computation Lab:
Consider a randomized pair design with $n$ units where two treatments are randomly assigned to each unit, resulting in a pair of observations $\left(X_{i}, Y_{i}\right)$, for $i=1, \ldots, n$ on each
Suppose that two drugs A and B are to be tested on 12 subjects' eyes. The drugs will be randomly assigned to the left eye or right eye based on the flip of a fair coin. If the coin toss is heads then
Consider the power function of one-sample z-test shown in Equation (4.3). What is the limit of the power function as $n \rightarrow \infty$ ? How about when $\mu_{1} \rightarrow \mu_{0}$ ? What do
Show that\[\begin{aligned}& P_{H_{1}}\left(\frac{\bar{X}-\mu_{0}}{\frac{S}{\sqrt{n}}} \geq t_{n-1,1-\alpha / 2}\right)+P_{H_{1}}\left(\frac{\bar{X}-\mu_{0}}{\frac{S}{\sqrt{n}}}
Derive equation (4.1) for the total sample of the two-sample t-test.Data from Equation 4.1 n = 2 40 (21-a/2 + 21-3)
Consider the derivation of Equation (4.5) for the sample size of a two-sample z-test with known variance and equal allocation.a. Identify terms in Equation (4.2) that are smaller than $\alpha / 2$
Consider Figure 4.2 and the $\mathrm{R}$ code that generated the plots. a. Modify the code so that sample size is on the $\mathrm{x}$-axis, and three different lines show the relationships between
Consider the plot of power as a function of effect size for the two-sample t-test shown in Figure 4.3.a. Create a plot to show power as a function of both effect size and sample size while keeping
Consider the plot of power as a function of the allocation ratio, $r$, for the two-sample z-test shown in Figure 4.4 and the $\mathrm{R}$ code for generating plot.a. Recall that $\mathrm{n}$ is the
The R function size2z.test () shown below implements the sample size formula for calculating the sample size for a test of $H_{0}: \theta=0$ versus $H_{1}: \theta eq 0$, where
A statistician is designing a phase III clinical trial comparing a continuous outcome in two groups receiving experimental versus standard therapy with a total sample size of 168 patients. The team
Let $X_{1}, X_{2}, \ldots, X_{n}$ be iid $N\left(\mu, \sigma^{2}\right)$.a. Show that the power function of the test $H_{0}: \mu=0$ versus $H_{1}: \mu>0$ at $\mu=1$
Use R to create a randomization scheme to randomize 222 subjects to three treatments such that there are an equal number of subjects assigned to each treatment. You may use the functions developed in
Explain when it is appropriate to use a randomized block design.
Show that\[S S_{T}=S S_{\text {Treat }}+S S_{E},\]where $S S_{T}=\sum_{i=1}^{k} \sum_{j=1}^{n_{i}}\left(y_{i j}-\bar{y} .\right)^{2}, S S_{\text {Treat }}=\sum_{i=1}^{k}
Consider the statistical model defined by Equation (5.1) and suppose $H_{0}: \tau_{1}=\cdots=\tau_{k}=0$ is true. Show that the following are true.a. $S S_{\text {Treat }} / \sigma^{2} \sim
Explain why the sample variance formula is\[\sum_{i=1}^{N} \frac{\left(y_{i}-\bar{y}\right)^{2}}{N-1}\]instead of\[\sum_{i=1}^{N} \frac{\left(y_{i}-\bar{y}\right)^{2}}{N}\]
The following data are the response times (in minutes) of six people measured after two treatments. The order in which each person received the treatments was determined by randomization. The
Let $y_{i j}$ for $i=1, \ldots, b$ and $j=1, \ldots, k$ be the measurement for the unit assigned to the $j^{\text {th }}$ treatment in the $i^{t h}$ block.a. Show that\[S S_{T}=S S_{\text {Treat }}+S
Consider the fisher. barley data frame from Example 5.3.a. Use $\mathrm{R}$ to compute $S S_{E}$. What are the degrees of freedom for $S S_{E}$ ?b. A statistician analyzing fisher.barley forgot to
Consider a randomized block design with $k$ treatments and $b$ blocks.a. Derive the least squares estimators of the treatment effects using i) dummy coding and ii) deviation coding.b. Verify the
As the statistician on a multidisciplinary research team, you are asked to create a randomization scheme to assign three treatments A, B, and $\mathrm{C}$, to 150 units in blocks of 6 . The scheme
Randomly permute each row of the matrix below using R.\[\left[\begin{array}{llll}1 & 1 & 1 & 1 \\2 & 2 & 2 & 2 \\3 & 3 & 3 & 3 \\4 & 4 & 4 & 4\end{array}\right]\]You can construct the matrix below
Consider the $4 \mathrm{x} 4$ square matrix below.\[\left[\begin{array}{llll}A & B & C & D \\D & A & B & C \\C & D & A & B \\B & C & D & A\end{array}\right]\]Suppose the rows correspond to subjects
Consider the Latin Square design and data from Bliss and Rose [1940] shown in Table 5.9.a. Use R to compute the ANOVA table. Interpret the results.b. Estimate the treatment effects. Interpret the
In questions 9, suppose the marketing manager is interested in the total number of trips to the shopping center in a given month, not in the average number of trips per person for the month.(a)
A consultant argues that location, the most important factor in the real estate business, was not considered in the test performed in question?
Use the information given in question 43 to randomly select five samples of four people and determine the mean and standard deviation for each sample.Question 43Consider the members of a group with

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