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 Correct Calculation And Interpretation Of The Standard Deviation Of The

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Introduction

In this article, we will explore the concept of standard deviation in the context of a real-world scenario. We will calculate and interpret the standard deviation of the proportion of twins born in a town, given a specific proportion and sample size.

The Problem

Suppose we randomly select 100 women from a town where the proportion of twins born is p=0.12p = 0.12. We want to calculate and interpret the standard deviation of the proportion of twins born in this sample.

Calculating the Standard Deviation

The standard deviation of a proportion is given by the formula:

σ=p(1−p)n\sigma = \sqrt{\frac{p(1-p)}{n}}

where pp is the population proportion, nn is the sample size, and σ\sigma is the standard deviation.

In this case, we have:

  • p=0.12p = 0.12
  • n=100n = 100

Plugging these values into the formula, we get:

σ=0.12(1−0.12)100\sigma = \sqrt{\frac{0.12(1-0.12)}{100}}

σ=0.12×0.88100\sigma = \sqrt{\frac{0.12 \times 0.88}{100}}

σ=0.1056100\sigma = \sqrt{\frac{0.1056}{100}}

σ=0.001056\sigma = \sqrt{0.001056}

σ=0.0324\sigma = 0.0324

Interpreting the Standard Deviation

The standard deviation of the proportion of twins born in this sample is 0.0324. This means that if we were to repeat this sampling process many times, we would expect the proportion of twins born in each sample to vary by approximately 0.0324 from the true population proportion of 0.12.

What Does This Mean?

In practical terms, this means that if we were to select 100 women from this town and calculate the proportion of twins born, we would expect the result to be close to 0.12, but not exactly 0.12. The standard deviation of 0.0324 gives us an idea of how much variation we would expect to see in the results.

Example

To illustrate this, let's consider an example. Suppose we select 100 women from this town and calculate the proportion of twins born. We get a result of 0.14. This is 0.02 above the true population proportion of 0.12. This is within one standard deviation of the true population proportion, which is a reasonable amount of variation.

Conclusion

In conclusion, we have calculated and interpreted the standard deviation of the proportion of twins born in a town, given a specific proportion and sample size. The standard deviation gives us an idea of how much variation we would expect to see in the results, and can be used to make inferences about the true population proportion.

Key Takeaways

  • The standard deviation of a proportion is given by the formula σ=p(1−p)n\sigma = \sqrt{\frac{p(1-p)}{n}}.
  • The standard deviation gives us an idea of how much variation we would expect to see in the results.
  • The standard deviation can be used to make inferences about the true population proportion.

References

  • [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Agresti, A., & Franklin, C. (2013). Statistics: The art and science of learning from data. Pearson Education.
    The Proportion of Twins Born in a Town: Calculating and Interpreting the Standard Deviation ====================================================================================

Q&A: Frequently Asked Questions

Q: What is the standard deviation of a proportion?

A: The standard deviation of a proportion is a measure of the amount of variation or dispersion in a set of data. It is calculated using the formula σ=p(1−p)n\sigma = \sqrt{\frac{p(1-p)}{n}}, where pp is the population proportion, nn is the sample size, and σ\sigma is the standard deviation.

Q: How do I calculate the standard deviation of a proportion?

A: To calculate the standard deviation of a proportion, you need to know the population proportion (pp) and the sample size (nn). You can then plug these values into the formula σ=p(1−p)n\sigma = \sqrt{\frac{p(1-p)}{n}} to get the standard deviation.

Q: What does the standard deviation of a proportion tell me?

A: The standard deviation of a proportion tells you how much variation you would expect to see in the results if you were to repeat the sampling process many times. It gives you an idea of how much the results may vary from the true population proportion.

Q: How do I interpret the standard deviation of a proportion?

A: To interpret the standard deviation of a proportion, you need to consider the context of the problem. For example, if the standard deviation is 0.05, this means that you would expect the results to vary by approximately 0.05 from the true population proportion. This can help you make inferences about the true population proportion.

Q: What is the difference between the standard deviation of a proportion and the standard error of a proportion?

A: The standard deviation of a proportion and the standard error of a proportion are related but distinct concepts. The standard deviation of a proportion is a measure of the amount of variation in a set of data, while the standard error of a proportion is a measure of the variability of the sample proportion from the true population proportion.

Q: How do I calculate the standard error of a proportion?

A: To calculate the standard error of a proportion, you need to know the sample proportion (p^\hat{p}) and the sample size (nn). You can then use the formula SE=p^(1−p^)nSE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} to get the standard error.

Q: What is the relationship between the standard deviation of a proportion and the standard error of a proportion?

A: The standard deviation of a proportion and the standard error of a proportion are related by the formula SE=σnSE = \frac{\sigma}{\sqrt{n}}. This means that the standard error of a proportion is equal to the standard deviation of a proportion divided by the square root of the sample size.

Q: How do I use the standard deviation of a proportion to make inferences about the true population proportion?

A: To use the standard deviation of a proportion to make inferences about the true population proportion, you need to consider the context of the problem and the results of the sampling process. For example, if the standard deviation is 0.05 and the sample proportion is 0.12, you may infer that the true population proportion is likely to be close to 0.12, but not exactly 0.12.

Conclusion

In conclusion, the standard deviation of a proportion is a useful tool for understanding the amount of variation in a set of data. By calculating and interpreting the standard deviation of a proportion, you can make inferences about the true population proportion and gain a deeper understanding of the data.

Key Takeaways

  • The standard deviation of a proportion is a measure of the amount of variation in a set of data.
  • The standard deviation of a proportion can be calculated using the formula σ=p(1−p)n\sigma = \sqrt{\frac{p(1-p)}{n}}.
  • The standard deviation of a proportion gives you an idea of how much variation you would expect to see in the results if you were to repeat the sampling process many times.
  • The standard error of a proportion is a measure of the variability of the sample proportion from the true population proportion.
  • The standard error of a proportion can be calculated using the formula SE=p^(1−p^)nSE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.
  • The standard deviation of a proportion and the standard error of a proportion are related by the formula SE=σnSE = \frac{\sigma}{\sqrt{n}}.