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The following problem is based on information from the National Oceanic and Atmospberic Administration (NOAA) Environmental Data Service. Let x be a random variable that represents the average daily temperature (in degrees Fahrenheit) in July in the town of Kit Carson, Colorado. The x distribution has a mean $\mu$ of approximately $75^{\circ} \mathrm{F}$ and standard deviation $\sigma$ of approximately $8^{\circ} \mathrm{F}$. A 20 -year study (620 July days) gave the entries in the rightmost column of the following table. $$ \scriptscriptstyle\begin{array}{lccc} \hline I & II & III & I\mathrm{V} \\ \hline & & & \text { Observed } \\ \text { Region under } & & \text { Expected % from } & \text { Number of Days } \\ \text { Normal Curve } & x^{\circ} \mathrm{F} & \text { Normal Curve } & \text { in 20 Years } \\ \hline \mu-3 \sigma \leq x<\mu-2 \sigma & 51 \leq x<59 & 2.35 \% & 16 \\ \mu-2 \sigma \leq x<\mu-\sigma & 59 \leq x<67 & 13.5 \% & 78 \\ \mu-\sigma \leq x<\mu & 67 \leq x<75 & 34 \% & 212 \\ \mu \leq x<\mu+\sigma & 75 \leq x<83 & 34 \% & 221 \\ \mu+\sigma \leq x<\mu+2 \sigma & 83 \leq x<91 & 13.5 \% & 81 \\ \mu+2 \sigma \leq x<\mu+3 \sigma & 91 \leq x<99 & 2.35 \% & 12 \\ \hline \end{array} $$ (i) Remember that $\mu=75$ and $\sigma=8.$. Write a brief explanation for Columns I, II, and III in the context of this problem. (ii) Use a 1% level of significance to test the claim that the average daily July temperature follows a normal distribution with $\mu=75$ and $\sigma=8$. What is the level of significance? State the null and alternate hypotheses.

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

Diamond Industries, Inc., began business on January 2, 2011. 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 2012, before the Wage and Tax Statements (Form W-2) could be prepared for distribution to employees and for filing with the Social Security Administration, the employees’ earnings records were inadvertently destroyed.

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% and Medicare tax at the rate of 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:

 Employee  Date First  Employed  Monthly  Salary  Monthly  Income Tax  Withheld  Beltran  Jan. 1 $4,300$681 Jeter  Apr. 1611,0002,508 Lee  Aug. 1 7,8001,612 Rodriguez  Nov. 163,000356 Santana  Mar. 16,1201,145 Ramirez  May 163,840566 Ordonez  Dec. 14,000606\begin{array}{lccc} \text { Employee } & \begin{array}{c} \text { Date First } \\ \text { Employed } \end{array} & \begin{array}{c} \text { Monthly } \\ \text { Salary } \end{array} & \begin{array}{c} \text { Monthly } \\ \text { Income Tax } \\ \text { Withheld } \end{array} \\ \hline \text { Beltran } & \text { Jan. 1 } & \$ 4,300 & \$ 681 \\ \text { Jeter } & \text { Apr. } 16 & 11,000 & 2,508 \\ \text { Lee } & \text { Aug. 1 } & 7,800 & 1,612 \\ \text { Rodriguez } & \text { Nov. } 16 & 3,000 & 356 \\ \text { Santana } & \text { Mar. } 1 & 6,120 & 1,145 \\ \text { Ramirez } & \text { May } 16 & 3,840 & 566 \\ \text { Ordonez } & \text { Dec. } 1 & 4,000 & 606 \end{array}

Instructions

  1. Calculate the amounts to be reported on each employee’s Wage and Tax Statement (Form W-2) for 2011.
  2. Calculate the following employer payroll taxes for the year: (a) social security; (b) Medicare; (c) state unemployment compensation at 4.4% on the first $9,000 of each employee’s earnings; (d) federal unemployment compensation at 0.8% on the first$9,000 of each employee’s earnings; (e) total.

Solution

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In this problem, we are tasked to calculate the amounts to be reported on Wage and Tax Statement Form W-2) and the employer's payroll taxes for the year.

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