To the Internal Revenue Service, the reasonableness of total itemized deductions depends on the taxpayer’s adjusted gross income. Large deductions, which include charity and medical deductions, are more reasonable for taxpayers with large adjusted gross incomes. If a taxpayer claims larger than average itemized deductions for a given level of income, the chances of an IRS audit are increased. Data (in thousands of dollars) on adjusted gross income and the average or reasonable amount of itemized deductions follow.

$$\begin{matrix} \text{ } & \text{Reasonable Amount of}\\ \text{Adjusted Gross Income (\$1000s)} & \text{Itemized Deductions (\$1000s)}\\ \hline \text{22} & \text{9.6}\\ \text{27} & \text{9.6}\\ \text{32} & \text{10.1}\\ \text{48} & \text{11.1}\\ \text{65} & \text{13.5}\\ \text{85} & \text{17.7}\\ \text{120} & \text{25.5}\\ \end{matrix}$$

Develop a scatter diagram for these data with adjusted gross income as the independent variable.

To the Internal Revenue Service, the reasonableness of total itemized deductions depends on the taxpayer’s adjusted gross income. Large deductions, which include charity and medical deductions, are more reasonable for taxpayers with large adjusted gross incomes. If a taxpayer claims larger than average itemized deductions for a given level of income, the chances of an IRS audit are increased. Data (in thousands of dollars) on adjusted gross income and the average or reasonable amount of itemized deductions follow.

$$\begin{matrix} \text{ } & \text{Reasonable Amount of}\\ \text{Adjusted Gross Income (\$1000s)} & \text{Itemized Deductions (\$1000s)}\\ \hline \text{22} & \text{9.6}\\ \text{27} & \text{9.6}\\ \text{32} & \text{10.1}\\ \text{48} & \text{11.1}\\ \text{65} & \text{13.5}\\ \text{85} & \text{17.7}\\ \text{120} & \text{25.5}\\ \end{matrix}$$

Estimate a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $52,500. If this taxpayer claimed itemized deductions of$20,400, would the IRS agent’s request for an audit appear justified? Explain.

An important application of regression analysis in accounting is in the estimation of cost. By collecting data on volume and cost and using the least squares method to develop an estimated regression equation relating volume and cost, an accountant can estimate the cost associated with a particular manufacturing volume. Consider the following sample of production volumes and total cost data for a manufacturing operation.

$$\begin{matrix} \text{Production Volume (units)} & \text{Total Cost (\$)}\\ \text{400} & \text{4000}\\ \text{450} & \text{5000}\\ \text{550} & \text{5400}\\ \text{600} & \text{5900}\\ \text{700} & \text{6400}\\ \text{750} & \text{7000}\\ \end{matrix}$$

Use these data to develop an estimated regression equation that could be used to predict the total cost for a given production volume.

In the earlier exercise, data were given on the adjusted gross income $x$ and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation $\hat{y}= 4.68 + .16x$, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of 52,500 dollars is 13,080 dollars. $\newline$
b. Develop a 95% prediction interval estimate for the amount of total itemized deductions for a particular taxpayer with an adjusted gross income of 52,500 dollar. $\newline$ Use your answer to part (b) to give the IRS agent a guideline as to the amount of total itemized deductions a taxpayer with an adjusted gross income of 52,500 dollar should claim before an audit is recommended.

In the earlier exercise, data were given on the adjusted gross income $x$ and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation $\hat{y}= 4.68 + .16x$, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of 52,500 dollars is 13,080 dollars. $\newline$
Develop a 95% prediction interval estimate for the amount of total itemized deductions for a particular taxpayer with an adjusted gross income of 52,500 dollar.

Use the following data:

$$\begin{matrix} x_i& \text{2} & \text{6} & \text{9} & \text{13} & \text{20}\\ y_i & \text{7} & \text{18} & \text{9} & \text{26} & \text{23}\\ \end{matrix}$$

The estimated regression equation for these data is $\hat { y } = 7.6 + .9 x .$ What percentage of the total sum of squares can be accounted for by the estimated regression equation? What is the value of the sample correlation coefficient?

In the earlier exercise, data were given on the adjusted gross income $x$ and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation $\hat{y}= 4.68 + .16x$, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of 52,500 dollars is 13,080 dollars. $\newline$ Develop a $95\%$ confidence interval for the mean amount of total itemized deductions for all taxpayers with an adjusted gross income of 52,500 dollar.

Use the following data:

$$\begin{matrix} x_i& \text{3} & \text{12} & \text{6} & \text{20} & \text{14}\\ y_i & \text{55} & \text{40} & \text{55} & \text{10} & \text{15}\\ \end{matrix}$$

Estimate the standard deviation of $\hat { y } _ { \mathrm { p } }$ when $x = 8$ .

One year, an accountant prepared individual income tax returns for 439 clients. For these individual tax returns, completion of the following schedules was required:

$$\begin{array}{ll} & \text{ 298 required schedule A (itemized deductions) } \\ & \text{ 256 required schedule B (interest and ordinary dividends) } \\ & \text{ 167 required schedule C (business income) } \\ & \text{ 212 required schedules A and B } \\ & \text{ 87 required schedules A and C } \\ & \text{ 61 required schedules B and C } \\ & \text{ 54 required schedules A, B, and C } \\\\ (a)& \text{ Draw a Venn diagram to illustrate this situation. } \\ (b)& \text{ How many individual tax returns required only schedule A } \\ (c) & \text{ How many individual tax returns required only schedule B } \\ (d)& \text{ How many individual tax returns required only schedule C } \\ (e)& \text{ How many individual tax returns required schedules A or C but not B } \\ (f) & \text{ How many individual tax returns required schedules A and C but not B } \\ (g)& \text{ How many clients required none of these three schedules?} \end{array}$$

Concur Technologies, Inc., is a large expense-management company located in Redmond, Washington. The Wall Street Journal asked Concur to examine the data from 8.3 million expense reports to provide insights regarding business travel expenses. Their analysis of the data showed that New York was the most expensive city, with an average daily hotel room rate of $198 and an average amount spent on entertainment, including group meals and tickets for shows, sports, and other events, of$172. In comparison, the U.S. averages for these two categories were $89 for the room rate and$99 for entertainment. The following table shows the average daily hotel room rate and the amount spent on entertainment for a random sample of 9 of the 25 most visited U.S. cities (The Wall Street Journal, August 18, 2011).

a. Develop a scatter diagram for these data with the room rate as the independent variable.

b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?

c. Develop the least squares estimated regression equation.

d. Provide an interpretation for the slope of the estimated regression equation.

e. The average room rate in Chicago is $\$128$, considerably higher than the U.S. average. Predict the entertainment expense per day for Chicago.

In the earlier exercise, data were given on the adjusted gross income $x$ and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation $\hat{y}= 4.68 + .16x$, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of 52,500 dollars is 13,080 dollars. $\newline$
If the particular taxpayer referred to in part (b) claimed total itemized deductions of 20,400 dollar, would the IRS agent’s request for an audit appear to be justified?

The Toyota Camry is one of the best-selling cars in North America. The cost of a previously owned Camry depends on many factors, including the model year, mileage, and condition. To investigate the relationship between the car’s mileage and the sales price for Camrys, the following data show the mileage and sale price for 19 sales (PriceHub Web site, February 24, 2012).

$$\begin{array}{cc} \text { Miles (1000s) } & \text{ Price (\$1,000s) } \\ 22 & 16.2 \\ 29 & 16.0 \\ 36 & 13.8 \\ 47 & 11.5 \\ 63 & 12.5 \\ 77 & 12.9 \\ 73 & 11.2 \\ 87 & 13.0 \\ 92 & 11.8 \\ 101 & 10.8 \\ 110 & 8.3 \\ 28 & 12.5 \\ 59 & 11.1 \\ 68 & 15.0 \\ 68 & 12.2 \\ 91 & 13.0 \\ 42 & 15.6 \\ 65 & 12.7 \\ 110 & 8.3 \end{array}$$

Test whether each of the regression parameters $\beta_0$ and $\beta_1$ is equal to zero at a $0.01$ level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?

Jensen Tire & Auto is in the process of deciding whether to purchase a maintenance contract for its new computer wheel alignment and balancing machine. Managers feel that maintenance expense should be related to usage, and they collected the following information on weekly usage (hours) and annual maintenance expense (in hundreds of dollars).

$$\begin{matrix} \text{Weekly Usage} & \text{Annual}\\ \text{(hours)} & \text{Maintenance Expense}\\ \hline \text{13} & \text{17.0}\\ \text{10} & \text{22.0}\\ \text{20} & \text{30.0}\\ \text{28} & \text{37.0}\\ \text{32} & \text{47.0}\\ \text{17} & \text{30.5}\\ \text{24} & \text{32.5}\\ \text{31} & \text{39.0}\\ \text{40} & \text{51.5}\\ \text{38} & \text{40.0}\\ \end{matrix}$$

Develop the estimated regression equation that relates annual maintenance expense to weekly usage.

The percent p of United States residents who were foreign born, or born outside of the United States, during the period 1910–2000 can be modeled by the equation p = 0.165| t - 60 | + 4.8 where t is the number of years since 1910. a. During the period 1910–2000, in approximately what year did foreign-born residents account for 13% of all residents? b. If the model holds for years after 2000, predict the year in which foreign-born residents will again account for 13% of all residents. c. According to the model, did foreign-born residents account for 4% of all residents at any time during the period 1910–2000? Explain your answer.