Related questions with answers

To obtain a more accurate picture of the impact of air bags on preventing deaths, it is necessary to account for the effect of occupants using both seatbelts and air bags. If the occupants of the vehicles in which air bags are installed are more likely to be also wearing seat belts, then it is possible that some of the apparent effectiveness of the air bags is in fact due to the increased usage of seat belts. Thus, one more 2 2 table is necessary, the table displaying a comparison of proper seat belt usage for occupants with air bags available with those for occupants without air bags available. That data are given here. a. Is there significant evidence of an association between air bag installation and the proper usage of seat belts? Use α\alpha = .05 b. Provide justification for your results in part (a). 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}

Air BagsSeat Belt Usage (Sealt Belt)Seat Belt Usage (No Sealt Belt)TotalYes4,871,940870,8755,742,815No2,902,6941,952,2114,854,905Total7,774,6342,823,08610,597,720\begin{matrix} \text{Air Bags} & \text{Seat Belt Usage (Sealt Belt)} & \text{Seat Belt Usage (No Sealt Belt)} & \text{Total}\\ \text{Yes} & \text{4,871,940} & \text{870,875} & \text{5,742,815}\\ \text{No} & \text{2,902,694} & \text{1,952,211} & \text{4,854,905}\\ \text{Total} & \text{7,774,634} & \text{2,823,086} & \text{10,597,720}\\ \end{matrix}

Question

A randomly chosen subject arrives for a study of exercise and fitness. Describe a sample space for this case. You measure the maximum heart rate (beats per minute).

Solution

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The sample space contains all possible outcomes for any subject/individual.

The maximum heart rate is 220 decreased by the age (and take the maximum age at about 120 years old).

S={ all numbers between 100 and 220 including}S=\{\text{ all numbers between 100 and 220 including}\}

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