A Poll Of 500 Maryland Adult Residents Was Conducted To Determine Whether They Favor The Widening Of Interstate 81. Suppose That, In Fact, 45% Of The Population Favors This Idea.a) What Is The Mean Of The Sampling Distribution Of { P $}$,

Mar 10, 2025 by ADMIN 239 views

1. A Poll of Maryland Adult Residents: Understanding the Sampling Distribution of p

1.1 Introduction

In statistics, a sampling distribution is a probability distribution of a statistic that is obtained from a large number of samples drawn from a population. In this case, we are interested in determining the mean of the sampling distribution of the proportion of Maryland adult residents who favor the widening of Interstate 81. To do this, we will use the results of a poll of 500 Maryland adult residents, which found that 45% of the population favors this idea.

1.2 The Sampling Distribution of p

The sampling distribution of p is a probability distribution of the sample proportion, where p is the population proportion. In this case, we are interested in finding the mean of the sampling distribution of p, denoted as μp.

1.3 The Formula for μp

The formula for the mean of the sampling distribution of p is given by:

μp = p

where p is the population proportion.

1.4 Applying the Formula to the Data

In this case, we are given that 45% of the population favors the widening of Interstate 81. Therefore, the population proportion p is equal to 0.45.

1.5 Calculating μp

Using the formula for μp, we can calculate the mean of the sampling distribution of p as follows:

μp = p = 0.45

1.6 Conclusion

In conclusion, the mean of the sampling distribution of p is equal to 0.45, which represents the population proportion of Maryland adult residents who favor the widening of Interstate 81.

1.7 Key Takeaways

The sampling distribution of p is a probability distribution of the sample proportion.
The mean of the sampling distribution of p is equal to the population proportion.
The formula for μp is given by μp = p.

1.8 References

[1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Larson, R. E., & Farber, M. (2018). Elementary statistics: Picturing the world. Cengage Learning.

1.9 Further Reading

For further reading on the topic of sampling distributions, we recommend the following resources:

[1] "Sampling Distribution" by Khan Academy
[2] "Sampling Distribution of the Mean" by Stat Trek

1.10 Final Thoughts

In this article, we have discussed the concept of the sampling distribution of p and how to calculate the mean of this distribution. We have also applied this concept to a real-world example involving a poll of Maryland adult residents. By understanding the sampling distribution of p, we can gain valuable insights into the population proportion and make informed decisions based on the data.

2.1 Introduction

In our previous article, we discussed the concept of the sampling distribution of p and how to calculate the mean of this distribution. In this article, we will answer some frequently asked questions (FAQs) related to the sampling distribution of p.

2.2 Q&A

2.2.1 Q: What is the difference between the population proportion and the sample proportion?

A: The population proportion (p) is the proportion of the population that has a certain characteristic, while the sample proportion (p̂) is the proportion of the sample that has that characteristic.

2.2.2 Q: How is the sampling distribution of p related to the population distribution?

A: The sampling distribution of p is a probability distribution of the sample proportion, which is based on the population distribution. The sampling distribution of p is a theoretical distribution that describes the possible values of the sample proportion.

2.2.3 Q: What is the formula for the standard deviation of the sampling distribution of p?

A: The formula for the standard deviation of the sampling distribution of p is given by:

σp = √(p(1-p)/n)

where p is the population proportion and n is the sample size.

2.2.4 Q: How does the sample size affect the sampling distribution of p?

A: The sample size affects the sampling distribution of p in that a larger sample size results in a more precise estimate of the population proportion.

2.2.5 Q: What is the relationship between the sampling distribution of p and the central limit theorem?

A: The central limit theorem states that the sampling distribution of p will be approximately normal for large sample sizes, regardless of the shape of the population distribution.

2.2.6 Q: How can we use the sampling distribution of p to make inferences about the population proportion?

A: We can use the sampling distribution of p to make inferences about the population proportion by calculating the sample proportion and using it to estimate the population proportion.

2.2.7 Q: What are some common applications of the sampling distribution of p?

A: The sampling distribution of p has many applications in statistics, including hypothesis testing, confidence intervals, and regression analysis.

2.3 Conclusion

In conclusion, the sampling distribution of p is a fundamental concept in statistics that describes the possible values of the sample proportion. By understanding the sampling distribution of p, we can make informed decisions based on the data and make inferences about the population proportion.

2.4 Key Takeaways

The sampling distribution of p is a probability distribution of the sample proportion.
The mean of the sampling distribution of p is equal to the population proportion.
The standard deviation of the sampling distribution of p is given by σp = √(p(1-p)/n).
The sample size affects the sampling distribution of p in that a larger sample size results in a more precise estimate of the population proportion.
The central limit theorem states that the sampling distribution of p will be approximately normal for large sample sizes.

2.5 References

[1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Larson, R. E., & Farber, M. (2018). Elementary statistics: Picturing the world. Cengage Learning.

2.6 Further Reading

For further reading on the topic of the sampling distribution of p, we recommend the following resources:

[1] "Sampling Distribution" by Khan Academy
[2] "Sampling Distribution of the Mean" by Stat Trek

2.7 Final Thoughts

In this article, we have answered some frequently asked questions related to the sampling distribution of p. By understanding the sampling distribution of p, we can make informed decisions based on the data and make inferences about the population proportion.