A Bicycle Club Conducted A Survey To Find Out Which City Bicycle Path Was Most Popular. Sixty Of The Club's 497 Members Were Chosen At Random And Asked Which Bike Path They Liked The Most. The Club Made A Table Of The

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Introduction

In statistics, a sampling distribution is a probability distribution of a statistic that is obtained from a large number of random samples from a given population. In this article, we will explore how a bicycle club conducted a survey to find out which city bicycle path was most popular among its members. The club randomly selected 60 of its 497 members and asked them to choose their favorite bike path. We will use this scenario to understand the concept of sampling distribution and how it can be used to make inferences about a population.

The Problem

The bicycle club has 497 members, and they want to know which city bicycle path is most popular among them. To do this, they randomly select 60 members and ask them to choose their favorite bike path. The club then creates a table to summarize the results of the survey.

The Data

Bike Path Number of Members
Path A 20
Path B 15
Path C 10
Path D 5
Path E 10

The Question

The club wants to know which bike path is most popular among its members. To answer this question, we need to calculate the proportion of members who chose each bike path.

Calculating the Proportion

To calculate the proportion of members who chose each bike path, we need to divide the number of members who chose each path by the total number of members surveyed (60).

Bike Path Proportion
Path A 20/60 = 0.33
Path B 15/60 = 0.25
Path C 10/60 = 0.17
Path D 5/60 = 0.08
Path E 10/60 = 0.17

Understanding the Sampling Distribution

The sampling distribution of a statistic is a probability distribution of that statistic that is obtained from a large number of random samples from a given population. In this case, the sampling distribution of the proportion of members who chose each bike path is a probability distribution of the proportions that we calculated earlier.

The Central Limit Theorem

The central limit theorem (CLT) states that the sampling distribution of the mean of a large sample of independent and identically distributed random variables will be approximately normally distributed, regardless of the shape of the population distribution. In this case, we are interested in the sampling distribution of the proportion of members who chose each bike path.

Applying the Central Limit Theorem

Since the CLT states that the sampling distribution of the mean of a large sample of independent and identically distributed random variables will be approximately normally distributed, we can apply this theorem to our data. We can calculate the mean and standard deviation of the proportions that we calculated earlier.

Calculating the Mean and Standard Deviation

To calculate the mean and standard deviation of the proportions, we need to use the following formulas:

Mean = (sum of proportions) / (number of proportions) Standard Deviation = sqrt((sum of squared differences from mean) / (number of proportions - 1))

Using these formulas, we can calculate the mean and standard deviation of the proportions.

Calculating the Mean

Mean = (0.33 + 0.25 + 0.17 + 0.08 + 0.17) / 5 Mean = 0.20

Calculating the Standard Deviation

Standard Deviation = sqrt((0.33 - 0.20)^2 + (0.25 - 0.20)^2 + (0.17 - 0.20)^2 + (0.08 - 0.20)^2 + (0.17 - 0.20)^2) / (5 - 1) Standard Deviation = sqrt(0.13 + 0.05 + 0.09 + 0.36 + 0.09) / 4 Standard Deviation = sqrt(0.72) / 4 Standard Deviation = 0.15

Understanding the Sampling Distribution

Now that we have calculated the mean and standard deviation of the proportions, we can understand the sampling distribution of the proportion of members who chose each bike path. The sampling distribution is a probability distribution of the proportions that we calculated earlier.

Interpreting the Results

Based on the sampling distribution, we can make inferences about the population. We can say that the proportion of members who chose each bike path is approximately normally distributed with a mean of 0.20 and a standard deviation of 0.15.

Conclusion

In this article, we explored how a bicycle club conducted a survey to find out which city bicycle path was most popular among its members. We used the concept of sampling distribution to understand the results of the survey and make inferences about the population. We calculated the proportion of members who chose each bike path and used the central limit theorem to understand the sampling distribution of the proportions. We also calculated the mean and standard deviation of the proportions and used them to make inferences about the population.

References

Further Reading

Introduction

In our previous article, we explored how a bicycle club conducted a survey to find out which city bicycle path was most popular among its members. We used the concept of sampling distribution to understand the results of the survey and make inferences about the population. In this article, we will answer some frequently asked questions about the concept of sampling distribution and its application in the bicycle club's survey.

Q&A

Q: What is a sampling distribution?

A: A sampling distribution is a probability distribution of a statistic that is obtained from a large number of random samples from a given population.

Q: Why is the sampling distribution important?

A: The sampling distribution is important because it allows us to make inferences about a population based on a sample of data. It helps us to understand the variability of the sample and to estimate the population parameters.

Q: What is the central limit theorem?

A: The central limit theorem (CLT) states that the sampling distribution of the mean of a large sample of independent and identically distributed random variables will be approximately normally distributed, regardless of the shape of the population distribution.

Q: How does the CLT apply to the bicycle club's survey?

A: In the bicycle club's survey, we calculated the proportion of members who chose each bike path. We then used the CLT to understand the sampling distribution of the proportions. The CLT helped us to estimate the population parameters and to make inferences about the population.

Q: What is the difference between a population and a sample?

A: A population is the entire group of individuals or items that we are interested in, while a sample is a subset of the population that we select for analysis.

Q: Why is it important to select a random sample?

A: It is important to select a random sample because it helps to ensure that the sample is representative of the population. This increases the accuracy of the results and allows us to make inferences about the population.

Q: How do we calculate the mean and standard deviation of a sampling distribution?

A: We calculate the mean and standard deviation of a sampling distribution by using the formulas:

Mean = (sum of proportions) / (number of proportions) Standard Deviation = sqrt((sum of squared differences from mean) / (number of proportions - 1))

Q: What is the significance of the mean and standard deviation in a sampling distribution?

A: The mean and standard deviation of a sampling distribution are important because they help us to understand the variability of the sample and to estimate the population parameters.

Q: Can we use the sampling distribution to make inferences about the population?

A: Yes, we can use the sampling distribution to make inferences about the population. By understanding the sampling distribution, we can estimate the population parameters and make predictions about the population.

Conclusion

In this article, we answered some frequently asked questions about the concept of sampling distribution and its application in the bicycle club's survey. We hope that this article has helped to clarify the concept of sampling distribution and its importance in making inferences about a population.

References

Further Reading