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CTU Industries, Inc., began business on January 22, 20092009. Salaries were paid to employees on the last day of each month, and social security tax, Medicare tax, and federal income tax were withheld in the required amounts. An employee who is hired in the middle of the month receives half the monthly salary for that month. All required payroll tax reports were filed, and the correct amount of payroll taxes was remitted by the company for the calendar year. Early in 20102010, before the Wage and Tax Statements (Form W-22) could be prepared for distribution to employees and for filing with the Social Security Administration, the employees’ earnings records were inadvertently destroyed.
\hspace{10pt} None of the employees resigned or were discharged during the year, and there were no changes in salary rates. The social security tax was withheld at the rate of 6.0%6.0\% on the first $100,000\$100,000 of salary and Medicare tax at the rate of 1.5%1.5\% on salary. Data on dates of employment, salary rates, and employees’ income taxes withheld, which are summarized as follows, were obtained from personnel records and payroll records:

MonthlyDate FirstMonthlyIncome TaxEmployeeEmployedSalaryWithheldBrownAug.1$3,600$552CarrollJan.29,5002,113GrobeMay16,5001,277MeyerJuly14,200702SabanJan.25,100927TresselApr. 163,200452WeisOct.13,000402\begin{array}{lccc} &&& \textbf{Monthly}\\ & \textbf{Date First} & \textbf{Monthly} & \textbf{Income Tax}\\ \textbf{Employee} & \textbf{Employed} & \textbf{Salary} & \textbf{Withheld}\\ \hline \text{Brown} & \text{Aug.}\hspace{9pt} 1 & \$3,600 & \$\hspace{13pt}552\\ \text{Carroll} & \text{Jan.}\hspace{9pt} 2 & \hspace{5pt} 9,500 & \hspace{5pt}2,113\\ \text{Grobe} & \text{May}\hspace{9pt} 1 & \hspace{5pt}6,500 & \hspace{5pt}1,277\\ \text{Meyer} & \text{July}\hspace{9pt} 1 & \hspace{5pt}4,200 & \hspace{15pt}702\\ \text{Saban} & \text{Jan.}\hspace{9pt} 2 & \hspace{5pt}5,100 & \hspace{15pt}927\\ \text{Tressel} & \text{Apr.}\ 16 & \hspace{5pt}3,200 & \hspace{15pt}452\\ \text{Weis} & \text{Oct.}\hspace{9pt} 1 & \hspace{5pt}3,000 & \hspace{15pt}402 \end{array}

Instructions\textsf{\textcolor{#4257b2}{Instructions}}
11. Calculate the amounts to be reported on each employee’s Wage and Tax Statement (Form W-22) for 2009,2009, arranging the data in the following form:

GrossFederal IncomeSocial SecurityMedicare TaxEmployeeEarningsTax WithheldTax WithheldWithheld\begin{array}{ccccc} & \textbf{Gross} & \textbf{Federal Income} & \textbf{Social Security} & \textbf{Medicare Tax}\\ \textbf{Employee} & \textbf{Earnings} & \textbf{Tax Withheld} & \textbf{Tax Withheld} & \textbf{Withheld}\\ \hline \end{array}

22. Calculate the following employer payroll taxes for the year: (a)\left(\text{a}\right) social security; (b)\left(\text{b}\right) Medicare; (c)\left(\text{c}\right) state unemployment compensation at 4.8%4.8\% on the first $10,000\$10,000 of each employee’s earnings; (d)\left(\text{d}\right) federal unemployment compensation at 0.8%0.8\% on the first $10,000\$10,000 of each employee’s earnings; (e)\left(\text{e}\right) total.

Which of the two safety devices appears to be more effective in preventing a death during an accident? Justify your answer using the information from the previous two exercises. REFERENCE: The authors also collected information about accidents concerning seat belt usage. The article compared fatality rates for occupants using seat belts properly with those for occupants not using seat belts. The data are given here. a. Calculate the odds of being killed in a harmful event car accident for a vehicle in which occupants were using seat belts and those who were not using seat belts. Interpret the two odds. b. Calculate the odds ratio of being killed in a harmful event car accident with and without seat belts being used properly. What does this ratio tell you about the importance of using seat belts? c. Is there significant evidence of a difference between vehicles with and without proper seat belt usage relative to the proportion of persons killed in a harmful event vehicle accident? Use α\alpha = .05. d. Place a 95% confidence interval on the odds ratio. Interpret this interval. 10.54 : The article “Who Wants Airbags” in Chance 18 (2005): 3–16 discusses whether air bags should be mandatory equipment in all new automobiles. Using data from the National Highway Traffic Safety Administration (NHTSA), they obtain the following information about fatalities and the usage of air bags and seat belts. All passenger cars sold in the U.S. starting in 1998 are required to have air bags. NHTSA estimates that air bags have saved 10,000 lives as of January 2004. The authors examined accidents in which there was a harmful event (personal or property), and from which at least one vehicle was towed. After some screening of the data, they obtained the following results. (The authors detail in their article the types of screening of the data that was done.) a. Calculate the odds of being killed in a harmful event car accident for a vehicle with and without air bags. Interpret the two odds. b. Calculate the odds ratio of being killed in a harmful event car accident with and without air bags. What does this ratio tell you about the importance of having air bags in a vehicle? c. Is there significant evidence of a difference between vehicles with and without air bags relative to the proportion of persons killed in a harmful event vehicle accident? Use α\alpha = .05. d. Place a 95% confidence interval on the odds ratio. Interpret this interval.

 Air Bag Installed YesAir Bag Installed NoTotalKilled19,27627,92447,200Survived5,723,5394,826,98210,550,521Total5,742,8154,854,90610,597,721\begin{matrix} \text{ } & \text{Air Bag Installed Yes} & \text{Air Bag Installed No} & \text{Total}\\ \text{Killed} & \text{19,276} & \text{27,924} & \text{47,200}\\ \text{Survived} & \text{5,723,539} & \text{4,826,982} & \text{10,550,521}\\ \text{Total} & \text{5,742,815} & \text{4,854,906} & \text{10,597,721}\\ \end{matrix}

 Seat Belt Usage (Sealt Belt)Seat Belt Usage (No Sealt Belt)TotalKilled16,00131,19947,200Survived7,758,6342,791,88710,550,521Total7,774,6352,823,08610,597,721\begin{matrix} \text{ } & \text{Seat Belt Usage (Sealt Belt)} & \text{Seat Belt Usage (No Sealt Belt)} & \text{Total}\\ \text{Killed} & \text{16,001} & \text{31,199} & \text{47,200}\\ \text{Survived} & \text{7,758,634} & \text{2,791,887} & \text{10,550,521}\\ \text{Total} & \text{7,774,635} & \text{2,823,086} & \text{10,597,721}\\ \end{matrix}

Question

The National Occupant Protection Use Survey (NOPUS) was conducted to provide probability-based data on motorcycle helmet use in the United States. The survey was conducted by sending observers to randomly selected roadway sites where they collected data on motorcycle helmet use, including the number of motorcyclists wearing a Department of Transportation (DOT)-compliant helmet (National Highway Traffic SafetyAdministration website, January 7, 2010). Sample data consistent with the most recent NOPUS are shown below.

Type of Helmet
RegionDOT-CompliantNoncompliant
Northeast9662
Midwest8643
South9249
West7616
Total350170

a. Use the sample data to compute an estimate of the probability that a motorcyclist wears a DOT-compliant helmet.

b. The probability that a motorcyclist wore a DOT-compliant helmet five years ago was .48, and last year this probability was .63. Would the National Highway Traffic Safety Administration be pleased with the most recent survey results?

c. What is the probability of DOT-compliant helmet use by region of the country? What region has the highest probability of DOT-compliant helmet use?

Solution

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In the following steps we will answer the questions (related to helmet use data) asked in (a) - (c) using the given table.

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