The Sampling Distribution of a Proportion

Oct 13, 2025 by windailys.com 4 views

Introduction

In statistics, the sampling distribution of a statistic is a probability distribution of a statistic that is obtained from a large number of samples drawn from a population. The shape of the sampling distribution is an important aspect of statistical inference, as it determines the properties of the statistic, such as its mean and standard deviation. In this article, we will discuss the shape of the sampling distribution of a proportion, given that the proportion of students at a large high school who live within five miles of the school is $p=0.19$ . We will also determine the best description of the shape for the sampling distribution of the proportion.

The Sampling Distribution of a Proportion

The sampling distribution of a proportion is a probability distribution of the proportion of successes in a sample of size $n$ . The mean of the sampling distribution of a proportion is equal to the population proportion, $p$ , and the standard deviation is given by $\sqrt{\frac{p(1-p)}{n}}$ . In this case, the population proportion is $p=0.19$ , and the sample size is $n=17$ .

Calculating the Standard Deviation

To calculate the standard deviation of the sampling distribution of the proportion, we can use the formula:

\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.19(1-0.19)}{17}}

Simplifying the expression, we get:

\sqrt{\frac{0.19(0.81)}{17}} = \sqrt{\frac{0.15419}{17}} = \sqrt{0.00908} = 0.0957

Determining the Shape of the Sampling Distribution

The shape of the sampling distribution of a proportion is determined by the sample size and the population proportion. If the sample size is large, the sampling distribution of the proportion is approximately normal, regardless of the population proportion. However, if the sample size is small, the sampling distribution of the proportion is approximately binomial.

In this case, the sample size is $n=17$ , which is relatively small. Therefore, the sampling distribution of the proportion is approximately binomial.

The Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with a constant probability of success. The probability mass function of the binomial distribution is given by:

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

where $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success.

The Shape of the Binomial Distribution

The shape of the binomial distribution is determined by the number of trials, $n$ , and the probability of success, $p$ . If $n$ is large and $p$ is close to 0 or 1, the binomial distribution is approximately normal. However, if $n$ is small or $p$ is close to 0.5, the binomial distribution is skewed.

In this case, the number of trials is $n=17$ , which is relatively small. Therefore, the binomial distribution is skewed.

Conclusion

In conclusion, the shape of the sampling distribution of the proportion is approximately binomial, given that the proportion of students at a large high school who live within five miles of the school is $p=0.19$ and the sample size is $n=17$ . The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with a constant probability of success. The shape of the binomial distribution is determined by the number of trials and the probability of success.

References

Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
DeGroot, M. H., & Schervish, M. J. (2012). Probability and statistics. Addison-Wesley.

Glossary

Sampling distribution: A probability distribution of a statistic that is obtained from a large number of samples drawn from a population.
Proportion: A measure of the number of successes in a sample, expressed as a fraction or percentage.
Binomial distribution: A discrete probability distribution that models the number of successes in a fixed number of independent trials, each with a constant probability of success.
The Proportion of Students at a Large High School: Understanding the Shape of the Sampling Distribution ===========================================================

Q&A: The Shape of the Sampling Distribution of a Proportion

Q: What is the shape of the sampling distribution of a proportion? A: The shape of the sampling distribution of a proportion is determined by the sample size and the population proportion. If the sample size is large, the sampling distribution of the proportion is approximately normal, regardless of the population proportion. However, if the sample size is small, the sampling distribution of the proportion is approximately binomial.

Q: What is the binomial distribution? A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with a constant probability of success. The probability mass function of the binomial distribution is given by:

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

where $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success.

Q: What is the shape of the binomial distribution? A: The shape of the binomial distribution is determined by the number of trials, $n$ , and the probability of success, $p$ . If $n$ is large and $p$ is close to 0 or 1, the binomial distribution is approximately normal. However, if $n$ is small or $p$ is close to 0.5, the binomial distribution is skewed.

Q: What is the standard deviation of the sampling distribution of a proportion? A: The standard deviation of the sampling distribution of a proportion is given by:

\sqrt{\frac{p(1-p)}{n}}

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

Q: How do I calculate the standard deviation of the sampling distribution of a proportion? A: To calculate the standard deviation of the sampling distribution of a proportion, you can use the formula:

\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.19(1-0.19)}{17}}

Simplifying the expression, you get:

\sqrt{\frac{0.19(0.81)}{17}} = \sqrt{\frac{0.15419}{17}} = \sqrt{0.00908} = 0.0957

Q: What is the best description of the shape for the sampling distribution of the proportion? A: The best description of the shape for the sampling distribution of the proportion is approximately binomial, given that the proportion of students at a large high school who live within five miles of the school is $p=0.19$ and the sample size is $n=17$ .

Q: What are the implications of the shape of the sampling distribution of a proportion? A: The shape of the sampling distribution of a proportion has important implications for statistical inference. If the sample size is large, the sampling distribution of the proportion is approximately normal, and we can use the central limit theorem to make inferences about the population proportion. However, if the sample size is small, the sampling distribution of the proportion is approximately binomial, and we need to use specialized methods to make inferences about the population proportion.

Q: What are some common applications of the sampling distribution of a proportion? A: The sampling distribution of a proportion has many common applications in statistics and research. Some examples include:

Estimating the proportion of a population that has a certain characteristic
Comparing the proportions of two or more populations
Testing hypotheses about the population proportion
Making predictions about the population proportion based on sample data

Q: What are some common mistakes to avoid when working with the sampling distribution of a proportion? A: Some common mistakes to avoid when working with the sampling distribution of a proportion include:

Assuming that the sampling distribution of a proportion is always normal, even when the sample size is small
Failing to account for the variability in the sampling distribution of a proportion
Using methods that are not appropriate for the shape of the sampling distribution of a proportion

Conclusion

In conclusion, the shape of the sampling distribution of a proportion is an important aspect of statistical inference. By understanding the shape of the sampling distribution of a proportion, we can make more accurate inferences about the population proportion and avoid common mistakes.