In A Survey Conducted At A Pet Store, 150 Customers Were Asked If They Owned Birds Or Fish. The Survey Data Are Shown In The Relative Frequency Table Below.$\[ \begin{tabular}{|c|c|c|c|} \hline & \text{Own A Bird} & \begin{tabular}{c} \text{Do

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Understanding Survey Data: A Mathematical Analysis of Bird and Fish Ownership

In a survey conducted at a pet store, 150 customers were asked if they owned birds or fish. The survey data are shown in the relative frequency table below. In this article, we will analyze the survey data using mathematical concepts and provide insights into the ownership of birds and fish among the surveyed customers.

Own a bird Do not own a bird Total
Own a fish 30 20 50
Do not own a fish 40 60 100
Total 70 80 150

Relative Frequency Table

Own a bird Do not own a bird Total
Own a fish 0.20 0.13 0.33
Do not own a fish 0.27 0.40 0.67
Total 0.47 0.53 1.00

Venn Diagrams

To better understand the survey data, we can use Venn diagrams to visualize the relationships between the two variables: owning a bird and owning a fish. A Venn diagram is a graphical representation of the relationships between sets.

# Create a Venn diagram
library(VennDiagram)
venn.diagram(c(30, 40), c(20, 60), category = c("Own a bird", "Do not own a bird"),
             main = "Venn Diagram of Bird and Fish Ownership")

The Venn diagram shows that 30 customers own both a bird and a fish, 40 customers own a bird but not a fish, and 20 customers own a fish but not a bird.

Conditional Probability

We can calculate the conditional probability of owning a bird given that a customer owns a fish. This is denoted as P(Bird|Fish) and can be calculated using the formula:

P(Bird|Fish) = P(Bird and Fish) / P(Fish)

Using the survey data, we can calculate the conditional probability as follows:

P(Bird|Fish) = 30 / 50 = 0.6

This means that if a customer owns a fish, there is a 60% chance that they also own a bird.

Joint Probability

We can also calculate the joint probability of owning a bird and a fish. This is denoted as P(Bird and Fish) and can be calculated using the formula:

P(Bird and Fish) = P(Bird) * P(Fish)

Using the survey data, we can calculate the joint probability as follows:

P(Bird and Fish) = 0.47 * 0.33 = 0.155

This means that there is a 15.5% chance that a customer owns both a bird and a fish.

Marginal Probability

We can also calculate the marginal probability of owning a bird. This is denoted as P(Bird) and can be calculated using the formula:

P(Bird) = P(Bird and Fish) + P(Bird and not Fish)

Using the survey data, we can calculate the marginal probability as follows:

P(Bird) = 0.155 + 0.27 = 0.425

This means that there is a 42.5% chance that a customer owns a bird.

In this article, we analyzed the survey data using mathematical concepts and provided insights into the ownership of birds and fish among the surveyed customers. We used Venn diagrams to visualize the relationships between the two variables, calculated conditional probabilities to determine the likelihood of owning a bird given that a customer owns a fish, and calculated joint and marginal probabilities to determine the likelihood of owning a bird and a fish. These mathematical concepts can be applied to various real-world scenarios to gain a deeper understanding of complex data.

Based on the analysis, we can make the following recommendations:

  • The pet store can target customers who own both a bird and a fish with special promotions and discounts to increase sales.
  • The pet store can also target customers who own a bird but not a fish with promotions and discounts on bird-related products.
  • The pet store can also target customers who own a fish but not a bird with promotions and discounts on fish-related products.

By applying these recommendations, the pet store can increase sales and improve customer satisfaction.

This analysis has several limitations. Firstly, the survey data may not be representative of the entire population of pet store customers. Secondly, the survey data may be subject to biases and errors. Finally, the analysis assumes that the relationships between the two variables are linear, which may not be the case in reality.

Future research can build on this analysis by:

  • Collecting more data on pet store customers to improve the accuracy of the analysis.
  • Using more advanced statistical techniques to analyze the data.
  • Exploring the relationships between other variables, such as age, income, and education level.

By addressing these limitations and exploring new research directions, we can gain a deeper understanding of the complex relationships between variables and make more informed decisions in various real-world scenarios.
Frequently Asked Questions: Understanding Survey Data on Bird and Fish Ownership

A: The purpose of the survey is to collect data on the ownership of birds and fish among pet store customers. The survey aims to provide insights into the relationships between the two variables and to identify potential target markets for the pet store.

A: The sample size of the survey is 150 customers. This sample size is sufficient to provide reliable estimates of the population parameters.

A: The relative frequency table is a table that shows the proportion of customers who own a bird, own a fish, or own both a bird and a fish. The table is used to visualize the relationships between the two variables.

A: The Venn diagram is a graphical representation of the relationships between the two variables. It shows the number of customers who own a bird, own a fish, or own both a bird and a fish.

A: The conditional probability is the probability of owning a bird given that a customer owns a fish. It is calculated using the formula P(Bird|Fish) = P(Bird and Fish) / P(Fish).

A: The joint probability is the probability of owning a bird and a fish. It is calculated using the formula P(Bird and Fish) = P(Bird) * P(Fish).

A: The marginal probability is the probability of owning a bird. It is calculated using the formula P(Bird) = P(Bird and Fish) + P(Bird and not Fish).

A: The recommendations based on the analysis are:

  • The pet store can target customers who own both a bird and a fish with special promotions and discounts to increase sales.
  • The pet store can also target customers who own a bird but not a fish with promotions and discounts on bird-related products.
  • The pet store can also target customers who own a fish but not a bird with promotions and discounts on fish-related products.

A: The limitations of the analysis are:

  • The survey data may not be representative of the entire population of pet store customers.
  • The survey data may be subject to biases and errors.
  • The analysis assumes that the relationships between the two variables are linear, which may not be the case in reality.

A: The future research directions are:

  • Collecting more data on pet store customers to improve the accuracy of the analysis.
  • Using more advanced statistical techniques to analyze the data.
  • Exploring the relationships between other variables, such as age, income, and education level.

A: The pet store can apply the recommendations by:

  • Targeting customers who own both a bird and a fish with special promotions and discounts.
  • Targeting customers who own a bird but not a fish with promotions and discounts on bird-related products.
  • Targeting customers who own a fish but not a bird with promotions and discounts on fish-related products.

By applying these recommendations, the pet store can increase sales and improve customer satisfaction.

A: The pet store can overcome the limitations of the analysis by:

  • Collecting more data on pet store customers to improve the accuracy of the analysis.
  • Using more advanced statistical techniques to analyze the data.
  • Exploring the relationships between other variables, such as age, income, and education level.

By addressing these limitations and exploring new research directions, the pet store can gain a deeper understanding of the complex relationships between variables and make more informed decisions in various real-world scenarios.