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The article "Checks Halt over 200,000 Gun Sales" (San Luis Obispo Tribune, June 5,2000 ) reported that required background checks blocked 204,000 gun sales in \(1999 .\) The article also indicated that state and local police reject a higher percentage of would-be gun buyers than does the FBI, stating, "The FBI performed 4.5 million of the 8.6 million checks, compared with 4.1 million by state and local agencies. The rejection rate among state and local agencies was \(3 \%,\) compared with \(1.8 \%\) for the FBI." a. Use the given information to estimate \(P(F), P(S)\), \(P(R \mid F),\) and \(P(R \mid S),\) where \(F=\) event that a randomly selected gun purchase background check is performed by the \(\mathrm{FBI}, S=\) event that a randomly selected gun purchase background check is performed by a state or local agency, and \(R=\) event that a randomly selected gun purchase background check results in a blocked sale. b. Use the probabilities from Part (a) to create a "hypothetical \(1000 "\) table. Use the table to calculate \(P(S \mid R),\) and write a sentence interpreting this value in the context of this problem.

Step by step solution

01

Estimation of probabilities

Firstly, probability that a randomly selected gun purchase background check is performed by the FBI (\(P(F)\)) would be the ratio of checks performed by the FBI to the total checks, which is \(4.5\) million / \(8.6\) million = \(0.523\). Similarly, the probability of a check being performed by a state or local agency (\(P(S)\)) is \(4.1\) million / \(8.6\) million = \(0.477\). \n\n Secondly, from the given problem, it is clear that rejection rate among FBI checks is \(1.8\%\) and state/local agency checks is \(3\%\). Hence, \(P(R|F) = 0.018\) and \(P(R|S) = 0.03\).

02

Creating hypothethical 1000 table

Using the probabilities calculated, we create a hypothetical 1000 table. It will look something like this: \n\n | Probabilities | Performed by FBI | Performed by state or local agency | Total | | ----------- | ----------- | ----------- | ----------- | | Randomly selected check resulted in blocked sale | 9.414 (0.018*523) | 14.31 (0.03*477) | 23.724 | | Otherwise | 513.586 | 462.69 | 976.276 | | Total | 523 (P(F)*1000) | 477 (P(S)*1000) | 1000 |

03

Calculation and interpretation of \(P(S|R)\)

This represents the probability that a background check is done by a state or local agency given that the check resulted in a blocked sale. This can be calculated as the number of blocks under state/local category divided by the total blocks. Hence, \(P(S | R) = 14.31 / 23.724 = 0.603\). Interpreting this, it says that more than 60% of the times a background check results in a blocked sale, it is likely that the check was performed by a state or local agency.

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Key Concepts

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

Conditional Probability

Conditional probability is a key concept in statistics that helps us understand the likelihood of an event occurring given another event already occurred. It's represented using the notation \(P(A \mid B)\), which reads as "the probability of event \(A\) given event \(B\)." In the context of the gun sales example, we use conditional probability to find the likelihood of a sale being blocked given the check was performed by the FBI or by state/local agencies. Conditional probability helps in narrowing down probable outcomes by considering existing conditions.

For instance, in the background check problem, \(P(R \mid F)\) represents the probability that a check results in a blocked sale given that the check was done by the FBI. Similarly, \(P(R \mid S)\) represents the probability of a blocked sale given the check was performed by state or local agencies. These probabilities reveal more about the process, helping us know which agencies are more stringent or effective at blocking sales.

Rejection Rate

Rejection rate is an essential metric in evaluating processes that involve approval or rejection of applications or requests. In this scenario, the rejection rate refers to the proportion of background checks on gun purchases that result in blocked sales.

- For the FBI, the rejection rate is \(1.8\%\). This indicates how strictly the FBI scrutinizes the applications.- For state and local agencies, the rate is \(3\%\), showing they have a higher tendency to reject gun purchase attempts compared to the FBI.

Understanding rejection rates is crucial in assessing the effectiveness of background check programs. A higher rejection rate might indicate a more rigorous check process, but it could also reflect different standards or criteria used by the agencies.

Statistical Estimation

Statistical estimation involves using data to determine values of certain parameters or probabilities. In the problem, we use statistical methods to estimate the probabilities of various events related to the background checks.

- \(P(F)\) and \(P(S)\) are probabilities that estimate how often the FBI or state/local agencies conduct checks respectively. We calculate these by dividing the number of checks each group conducts by the total number of checks.- The estimated probabilities for rejection \(P(R \mid F)\) and \(P(R \mid S)\) are based on the given rejection rates.- These estimates help us form a clearer picture of the process and develop informed expectations about the rejection behavior based on previous data.

Hypothetical Table

A hypothetical table, sometimes called a contingency table, is a useful tool in probability and statistics. It organizes data to help analyze relationships between different variables. In this problem, creating a hypothetical table lets us visually estimate probabilities and understand interactions between events.

We create one by deciding on a total, like 1000 checks, and use the estimated probabilities to allocate numbers:

• Perform checks by the FBI (\(P(F)\)) and by state/local agencies (\(P(S)\)).
• Calculate how many checks result in blocked sales based on the rejection rates \(P(R \mid F)\) and \(P(R \mid S)\).

This structured format helps simplify tasks of data analysis and probability computation by breaking down data into manageable segments.

Background Checks

Background checks in this context refer to the verification process of individuals attempting to purchase firearms. Authorities conduct these checks to ensure safety by rejecting applications that do not meet legal or safety standards.

The distinction between the FBI and local/state agency checks indicates different procedural practices or standards. Local and state agencies have a higher rejection rate, suggesting they might enforce stricter criteria or review standards compared to the FBI.

Understanding differences in background checks helps in interpreting rejection rates and informing policy-making to improve efficacy and safety in gun transactions.