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Chapter 6: Problem 35

The Cedar Rapids Gazette (November 20, 1999) reported the following information on compliance with child restraint laws for cities in Iowa: $$ \begin{array}{lcc} & \begin{array}{c} \text { Number of } \\ \text { Children } \\ \text { Observed } \end{array} & \begin{array}{c} \text { Number } \\ \text { Properly } \\ \text { City } \end{array} & \text { Restrained } \\ \hline \text { Cedar Falls } & 210 & 173 \\ \text { Cedar Rapids } & 231 & 206 \\ \text { Dubuque } & 182 & 135 \\ \text { Iowa City (city) } & 175 & 140 \\ \text { Iowa City (interstate) } & 63 & 47 \\ \hline \end{array} $$ a. Use the information provided to estimate the following probabilities: i. The probability that a randomly selected child is properly restrained given that the child is observed in Dubuque. ii. The probability that a randomly selected child is properly restrained given that the child is observed in a city that has "Cedar" in its name. b. Suppose that you are observing children in the Iowa City area. Use a tree diagram to illustrate the possible outcomes of an observation that considers both the location of the observation (city or interstate) and whether the child observed was properly restrained.

Short Answer

Expert verified
i. The probability that a randomly selected child is properly restrained in Dubuque is \( \frac{135}{182} \) ii. The probability that a randomly selected child is properly restrained in a city with 'Cedar' in its name is \( \frac{379}{441} \)

Step by step solution

01

Calculate the Probability for Dubuque

First, we calculate the probability of a child being restrained given they are in Dubuque. This is a case of conditional probability. The formula for calculating conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). In this scenario, Event A is a child being properly restrained and Event B is the child being observed in Dubuque. From the data provided, we can notice that out of 182 children observed in Dubuque, 135 were properly restrained. Hence, the conditional probability becomes \( P(A|B) = \frac{135}{182} \).
02

Calculate the Probability for Cities with 'Cedar' in Name

Next, we have to calculate the probability of a child being properly restrained if observed in a city with 'Cedar' in its name. This includes two cities - Cedar Falls and Cedar Rapids. So, we consider the total number of children and those restrained in both cities. The total number of children in Cedar Falls and Cedar Rapids is 210 + 231 = 441 and the number of children properly restrained in both cities is 173 + 206 = 379. Conditional probability will therefore be \( P(A|B) = \frac{379}{441} \).
03

Draw Tree Diagram for Iowa City Area

For Iowa city area, there are two observation locations - city and interstate, and two possible outcomes for each location - child properly restrained or not. You can visualize the same with a tree diagram having four distinct outcomes - (City, Restrained), (City, Not Restrained), (Interstate, Restrained), (Interstate, Not Restrained).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Estimation
Probability estimation is a fundamental aspect of statistics, allowing us to predict the likelihood of specific events occurring based on the available data. It's particularly crucial in the context of making informed decisions, such as understanding compliance rates of child restraint laws in different cities.

For instance, in the exercise given, the probability estimation involves determining how likely it is that a child is properly restrained in a vehicle, based on observations from various cities in Iowa. By using the provided data, we can calculate the conditional probability, which is the probability of an event occurring given that another event has already occurred.

In the scenario of Dubuque, the probability that a child is properly restrained, given that the observation occurred in Dubuque, is calculated using the ratio of the number of children properly restrained to the total number of children observed. This ratio provides a practical estimation of the probability of compliance with child restraint laws in Dubuque, offering valuable insight for policymakers and law enforcement agencies.

Similarly, the probability that a child observed in a city with 'Cedar' in its name is properly restrained can be estimated by combining the data for Cedar Falls and Cedar Rapids. By summing the total number of children observed and those properly restrained across both cities, we obtain a broader picture of compliance within these two regions.

These estimations are critical for understanding and improving public safety initiatives. They serve as a measure of how well the laws are being followed and the effectiveness of current enforcement strategies.
Tree Diagram
A tree diagram is a visual representation that clearly illustrates all possible outcomes of a scenario and their probabilities. It's a highly effective tool in probability estimation, especially when dealing with scenarios that have multiple stages or conditions.

In our exercise, we are asked to consider the Iowa City area where observations can occur in two distinct places鈥攚ithin the city or on the interstate. For each location, there are two outcomes of interest: whether the child is properly restrained or not. By drawing a tree diagram, we can organize this information in a way that is easy to interpret.

Steps to Create a Tree Diagram

  • Start by drawing a fork for the first decision point or condition: the location of the child (city or interstate).
  • From each branch of the first fork, draw additional forks representing the possible outcomes at that location (restrained or not restrained).
  • Finally, annotate the branches with probabilities or observed frequencies to complete the diagram.

The final tree diagram for the Iowa City area will have four end points, each corresponding to one of the possible outcomes within the given scenario. This visual allows for quick comprehension and is particularly helpful for identifying different pathways and their related probabilities.
Child Restraint Laws Compliance
Child restraint laws are regulations designed to ensure that children are securely fastened in suitable car seats while traveling in vehicles to reduce the risk of injury or death in case of an accident. Compliance with these laws is critical for child safety, and understanding the level of compliance can help in designing better public safety initiatives and educational programs.

In the exercise, compliance with child restraint laws is assessed by calculating the probability that a child was properly restrained based on observations from various cities. These probabilities serve as an estimate of compliance, revealing how frequently the laws are being followed. The estimated probabilities provide valuable feedback on where targeted awareness campaigns or stricter enforcement may be needed.

Monitoring and improving compliance with child restraint laws is vital, as it directly correlates with reducing child fatalities and injuries in automobile accidents. By using probability estimation and analyzing compliance rates in different regions, stakeholders can identify patterns and implement strategies to enhance the effectiveness of child safety regulations.

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Most popular questions from this chapter

Four students must work together on a group project. They decide that each will take responsibility for a particular part of the project, as follows: Because of the way the tasks have been divided, one student must finish before the next student can begin work. To ensure that the project is completed on time, a schedule is established, with a deadline for each team member. If any one of the team members is late, the timely completion of the project is jeopardized. Assume the following probabilities: 1\. The probability that Maria completes her part on time is \(.8\). 2\. If Maria completes her part on time, the probability that Alex completes on time is \(.9\), but if Maria is late, the probability that Alex completes on time is only . 6 . 3\. If Alex completes his part on time, the probability that Juan completes on time is \(.8\), but if Alex is late, the probability that Juan completes on time is only .5. 4\. If Juan completes his part on time, the probability that Jacob completes on time is \(.9\), but if Juan is late, the probability that Jacob completes on time is only \(.7 .\) Use simulation (with at least 20 trials) to estimate the probability that the project is completed on time. Think carefully about this one. For example, you might use a random digit to represent each part of the project (four in all). For the first digit (Maria's part), \(1-8\) could represent on time and 9 and 0 could represent late. Depending on what happened with Maria (late or on time), you would then look at the digit representing Alex's part. If Maria was on time, \(1-9\) would represent on time for Alex, but if Maria was late, only \(1-6\) would represent on time. The parts for Juan and Jacob could be handled similarly.

An article in the New York Times (March 2, 1994) reported that people who suffer cardiac arrest in New York City have only a 1 in 100 chance of survival. Using probability notation, an equivalent statement would be \(P\) (survival \()=.01\) for people who suffer a cardiac arrest in New York City. (The article attributed this poor survival rate to factors common in large cities: traffic congestion and the difficulty of finding victims in large buildings.) a. Give a relative frequency interpretation of the given probability. b. The research that was the basis for the New York Times article was a study of 2329 consecutive cardiac arrests in New York City. To justify the " 1 in 100 chance of survival" statement, how many of the 2329 cardiac arrest sufferers do you think survived? Explain.

The Associated Press (San Luis Obispo TelegramTribune, August 23,1995 ) reported on the results of mass screening of schoolchildren for tuberculosis (TB). For Santa Clara County, California, the proportion of all tested kindergartners who were found to have TB was .0006. The corresponding proportion for recent immigrants (thought to be a high-risk group) was .0075. Suppose that a Santa Clara County kindergartner is selected at random. Are the events selected student is a recent immigrant and selected student has \(T B\) independent or dependent events? Justify your answer using the given information.

In a small city, approximately \(15 \%\) of those eligible are called for jury duty in any one calendar year. People are selected for jury duty at random from those eligible, and the same individual cannot be called more than once in the same year. What is the probability that a particular eligible person in this city is selected two years in a row? three years in a row?

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