Use The Contingency Table Below To Estimate The Following Probability:P(Cecil Student Is Republican Or Christian) = Round Your Answer To Four Decimal Places. Example: 0.1911Contingency Table:| | Atheist / Agnostic | Catholic |

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Introduction

In probability theory, a contingency table is a table used to display the relationship between two categorical variables. It is a powerful tool for analyzing and understanding the relationships between different variables. In this article, we will use a contingency table to estimate the probability of a Cecil student being a Republican or Christian.

Understanding the Contingency Table

The contingency table provided is as follows:

Atheist / Agnostic Catholic
Republican 23 17
Christian 15 30
Not Republican/Christian 42 13

Defining the Problem

We are asked to estimate the probability of a Cecil student being a Republican or Christian. This means we need to find the probability of the union of the two events, P(R or C), where R represents the event of being a Republican and C represents the event of being a Christian.

Using the Contingency Table to Estimate the Probability

To estimate the probability of a Cecil student being a Republican or Christian, we need to use the following formula:

P(R or C) = P(R) + P(C) - P(R and C)

where P(R) is the probability of being a Republican, P(C) is the probability of being a Christian, and P(R and C) is the probability of being both a Republican and a Christian.

Calculating the Probabilities

From the contingency table, we can calculate the following probabilities:

  • P(R) = (Number of Republicans + Number of Christians) / Total number of students
  • P(C) = (Number of Republicans + Number of Christians) / Total number of students
  • P(R and C) = (Number of Republicans and Christians) / Total number of students

Using the contingency table, we can calculate the following values:

  • Number of Republicans = 23 + 17 = 40
  • Number of Christians = 15 + 30 = 45
  • Number of Republicans and Christians = 17 + 15 = 32
  • Total number of students = 23 + 42 + 17 + 13 + 15 + 30 = 140

Calculating the Probabilities (continued)

Now we can calculate the probabilities:

  • P(R) = (40 + 45) / 140 = 85 / 140 = 0.6071
  • P(C) = (40 + 45) / 140 = 85 / 140 = 0.6071
  • P(R and C) = 32 / 140 = 0.2286

Estimating the Probability of a Cecil Student Being a Republican or Christian

Now we can use the formula to estimate the probability of a Cecil student being a Republican or Christian:

P(R or C) = P(R) + P(C) - P(R and C) = 0.6071 + 0.6071 - 0.2286 = 0.9855

Rounding the Answer

We are asked to round the answer to four decimal places. Therefore, the estimated probability of a Cecil student being a Republican or Christian is:

P(R or C) = 0.9855

Conclusion

In this article, we used a contingency table to estimate the probability of a Cecil student being a Republican or Christian. We calculated the probabilities of being a Republican, being a Christian, and being both a Republican and a Christian. We then used the formula to estimate the probability of a Cecil student being a Republican or Christian. The estimated probability is 0.9855.

References

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Q: What is a contingency table?

A: A contingency table is a table used to display the relationship between two categorical variables. It is a powerful tool for analyzing and understanding the relationships between different variables.

Q: How do I use a contingency table to estimate the probability of an event?

A: To estimate the probability of an event using a contingency table, you need to follow these steps:

  1. Identify the event you want to estimate the probability of.
  2. Calculate the number of individuals who belong to the event.
  3. Calculate the total number of individuals in the population.
  4. Use the formula P(event) = (Number of individuals in the event) / (Total number of individuals) to estimate the probability.

Q: What is the difference between P(R) and P(C)?

A: P(R) and P(C) are the probabilities of being a Republican and being a Christian, respectively. They are calculated by dividing the number of individuals who belong to each category by the total number of individuals in the population.

Q: How do I calculate P(R and C)?

A: P(R and C) is the probability of being both a Republican and a Christian. It is calculated by dividing the number of individuals who belong to both categories by the total number of individuals in the population.

Q: What is the formula for estimating the probability of the union of two events?

A: The formula for estimating the probability of the union of two events is:

P(R or C) = P(R) + P(C) - P(R and C)

Q: Can I use a contingency table to estimate the probability of more than two events?

A: Yes, you can use a contingency table to estimate the probability of more than two events. However, the formula for estimating the probability of the union of multiple events becomes more complex.

Q: What are some common applications of contingency tables?

A: Contingency tables are commonly used in:

  • Marketing research to analyze customer behavior
  • Medical research to analyze disease outcomes
  • Social sciences to analyze demographic data
  • Business to analyze customer preferences

Q: How do I choose the right contingency table for my analysis?

A: To choose the right contingency table for your analysis, you need to consider the following factors:

  • The number of variables you want to analyze
  • The type of data you have (categorical or numerical)
  • The level of detail you want to achieve in your analysis

Q: Can I use a contingency table to analyze numerical data?

A: No, contingency tables are used to analyze categorical data. If you have numerical data, you may want to consider using other statistical tools, such as regression analysis or hypothesis testing.

Q: How do I interpret the results of a contingency table analysis?

A: To interpret the results of a contingency table analysis, you need to consider the following factors:

  • The size of the contingency table
  • The number of individuals in each category
  • The level of significance of the results

Q: What are some common pitfalls to avoid when using contingency tables?

A: Some common pitfalls to avoid when using contingency tables include:

  • Not accounting for confounding variables
  • Not using the correct statistical tests
  • Not interpreting the results correctly

Q: Can I use a contingency table to analyze data from a survey?

A: Yes, you can use a contingency table to analyze data from a survey. However, you need to consider the following factors:

  • The sample size of the survey
  • The level of detail in the survey questions
  • The type of data collected (categorical or numerical)