The Proportion Of Twins Born In A Town Is $p=0.12$. Suppose We Randomly Select 100 Women From This Town Who Give Birth In The Next Year. Which Of The Following Is The Mean Of The Sampling Distribution Of $\hat{p}$?A.

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Introduction

In statistics, the sampling distribution of a statistic is a probability distribution of the statistic that would result from an infinite number of samples of the same size from the same population. In this article, we will focus on the sampling distribution of the proportion of twins born in a town, denoted by $\hat{p}$. We will assume that the proportion of twins born in the town is $p=0.12$. Our goal is to determine the mean of the sampling distribution of $\hat{p}$ when we randomly select 100 women from this town who give birth in the next year.

The Sampling Distribution of $\hat{p}$

The sampling distribution of $\hat{p}$ is a probability distribution of the proportion of twins born in a sample of 100 women from the town. Since we are sampling without replacement, the sampling distribution of $\hat{p}$ is approximately normal with mean $p$ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.

The Mean of the Sampling Distribution of $\hat{p}$

The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$. In this case, the population proportion is $p=0.12$. Therefore, the mean of the sampling distribution of $\hat{p}$ is also 0.12.

Derivation of the Mean of the Sampling Distribution of $\hat{p}$

To derive the mean of the sampling distribution of $\hat{p}$, we can use the following formula:

$$\mu_{\hat{p}} = E(\hat{p}) = \frac{1}{n} \sum_{i=1}^{n} X_i$$

where $X_i$ is the number of twins born in the $i^{th}$ sample, and $n$ is the sample size.

Since the $X_i$'s are independent and identically distributed random variables with mean $np$ and variance $np(1-p)$, we can write:

$$E(\hat{p}) = \frac{1}{n} \sum_{i=1}^{n} E(X_i) = \frac{1}{n} \cdot np = p$$

Therefore, the mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$.

Conclusion

In conclusion, the mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$. In this case, the population proportion is $p=0.12$. Therefore, the mean of the sampling distribution of $\hat{p}$ is also 0.12. This result is consistent with the formula for the mean of the sampling distribution of $\hat{p}$, which is equal to the population proportion $p$.

References

• Casella, G., & Berger, R. L. (2002). Statistical inference. Duxbury.
• Hogg, R. V., & Tanis, E. A. (2001). Probability and statistical inference. Prentice Hall.

Mathematical Derivation

To derive the mean of the sampling distribution of $\hat{p}$, we can use the following formula:

$$\mu_{\hat{p}} = E(\hat{p}) = \frac{1}{n} \sum_{i=1}^{n} X_i$$

where $X_i$ is the number of twins born in the $i^{th}$ sample, and $n$ is the sample size.

Since the $X_i$'s are independent and identically distributed random variables with mean $np$ and variance $np(1-p)$, we can write:

$$E(\hat{p}) = \frac{1}{n} \sum_{i=1}^{n} E(X_i) = \frac{1}{n} \cdot np = p$$

Therefore, the mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$.

Sampling Distribution of $\hat{p}$

The sampling distribution of $\hat{p}$ is a probability distribution of the proportion of twins born in a sample of 100 women from the town. Since we are sampling without replacement, the sampling distribution of $\hat{p}$ is approximately normal with mean $p$ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.

Properties of the Sampling Distribution of $\hat{p}$

The sampling distribution of $\hat{p}$ has the following properties:

• The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$.
• The standard deviation of the sampling distribution of $\hat{p}$ is equal to $\sqrt{\frac{p(1-p)}{n}}$.
• The sampling distribution of $\hat{p}$ is approximately normal.

Real-World Applications

The sampling distribution of $\hat{p}$ has many real-world applications, including:

• Estimating the proportion of twins born in a town.
• Estimating the proportion of people with a certain characteristic in a population.
• Estimating the probability of a certain event occurring in a population.

Conclusion

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Q: What is the sampling distribution of $\hat{p}$?

A: The sampling distribution of $\hat{p}$ is a probability distribution of the proportion of twins born in a sample of 100 women from the town. Since we are sampling without replacement, the sampling distribution of $\hat{p}$ is approximately normal with mean $p$ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.

Q: What is the mean of the sampling distribution of $\hat{p}$?

A: The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$. In this case, the population proportion is $p=0.12$. Therefore, the mean of the sampling distribution of $\hat{p}$ is also 0.12.

Q: How do we derive the mean of the sampling distribution of $\hat{p}$?

A: To derive the mean of the sampling distribution of $\hat{p}$, we can use the following formula:

$$\mu_{\hat{p}} = E(\hat{p}) = \frac{1}{n} \sum_{i=1}^{n} X_i$$

where $X_i$ is the number of twins born in the $i^{th}$ sample, and $n$ is the sample size.

Since the $X_i$'s are independent and identically distributed random variables with mean $np$ and variance $np(1-p)$, we can write:

$$E(\hat{p}) = \frac{1}{n} \sum_{i=1}^{n} E(X_i) = \frac{1}{n} \cdot np = p$$

Therefore, the mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$.

Q: What are the properties of the sampling distribution of $\hat{p}$?

A: The sampling distribution of $\hat{p}$ has the following properties:

• The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$.
• The standard deviation of the sampling distribution of $\hat{p}$ is equal to $\sqrt{\frac{p(1-p)}{n}}$.
• The sampling distribution of $\hat{p}$ is approximately normal.

Q: What are the real-world applications of the sampling distribution of $\hat{p}$?

A: The sampling distribution of $\hat{p}$ has many real-world applications, including:

• Estimating the proportion of twins born in a town.
• Estimating the proportion of people with a certain characteristic in a population.
• Estimating the probability of a certain event occurring in a population.

Q: How do we use the sampling distribution of $\hat{p}$ in practice?

A: To use the sampling distribution of $\hat{p}$ in practice, we can follow these steps:

1. Collect a random sample of 100 women from the town.
2. Calculate the proportion of twins born in the sample.
3. Use the sampling distribution of $\hat{p}$ to estimate the population proportion $p$.
4. Use the estimated value of $p$ to make inferences about the population.

Q: What are the limitations of the sampling distribution of $\hat{p}$?

A: The sampling distribution of $\hat{p}$ has the following limitations:

• The sampling distribution of $\hat{p}$ is only approximately normal.
• The sampling distribution of $\hat{p}$ assumes that the sample is randomly selected from the population.
• The sampling distribution of $\hat{p}$ assumes that the sample size is large enough to be representative of the population.

Conclusion

In conclusion, the sampling distribution of $\hat{p}$ is a powerful tool for estimating the population proportion $p$. By understanding the properties and limitations of the sampling distribution of $\hat{p}$, we can use it to make informed decisions in a variety of real-world applications.