A One-sample Z-test For A Population Proportion Will Be Conducted Using A Simple Random Sample Selected Without Replacement From A Population. Which Of The Following Is A Check For Independence?A. $m_{0} \geq 10$ And $n \left( 1 - P_{0}

Introduction

In statistics, a one-sample z-test for a population proportion is a widely used method for determining whether a sample proportion is significantly different from a known population proportion. This test is particularly useful in scenarios where the sample size is large and the population size is finite. However, before conducting a one-sample z-test, it is essential to verify that the sample is independent of the population. In this article, we will discuss the check for independence in the context of a one-sample z-test for a population proportion.

What is Independence in Statistics?

In statistics, independence refers to the absence of any relationship between two or more variables. In the context of a one-sample z-test for a population proportion, independence is crucial because it ensures that the sample is not influenced by the population. If the sample is not independent of the population, the results of the test may be biased or unreliable.

Check for Independence: A. $m_{0} \geq 10$ and $n \left( 1 - p_{0} \right) \geq 10$

The check for independence in a one-sample z-test for a population proportion is given by the condition:

$$m_{0} \geq 10 \text{ and } n \left( 1 - p_{0} \right) \geq 10$$

where $m_{0}$ is the expected number of successes in the sample, $n$ is the sample size, and $p_{0}$ is the known population proportion.

Understanding the Condition

To understand the condition, let's break it down into two parts:

1. $m_{0} \geq 10$: This part of the condition ensures that the expected number of successes in the sample is at least 10. This is a reasonable assumption because if the expected number of successes is less than 10, the sample may not be representative of the population.
2. $n \left( 1 - p_{0} \right) \geq 10$: This part of the condition ensures that the expected number of failures in the sample is at least 10. This is also a reasonable assumption because if the expected number of failures is less than 10, the sample may not be representative of the population.

Why is this Condition Important?

The condition $m_{0} \geq 10$ and $n \left( 1 - p_{0} \right) \geq 10$ is important because it ensures that the sample is independent of the population. If the sample is not independent of the population, the results of the test may be biased or unreliable.

Example

Suppose we want to conduct a one-sample z-test for a population proportion to determine whether a sample of 100 students has a significantly different proportion of students who prefer a particular brand of coffee compared to a known population proportion of 0.6. The expected number of successes in the sample is $m_{0} = n p_{0} = 100 \times 0.6 = 60$. The expected number of failures in the sample is $n \left( 1 - p_{0} \right) = 100 \left( 1 - 0.6 \right) = 40$. Since both $m_{0}$ and $n \left( 1 - p_{0} \right)$ are greater than or equal to 10, the condition is satisfied, and we can proceed with the one-sample z-test.

Conclusion

In conclusion, the check for independence in a one-sample z-test for a population proportion is given by the condition $m_{0} \geq 10$ and $n \left( 1 - p_{0} \right) \geq 10$. This condition ensures that the sample is independent of the population, which is essential for obtaining reliable and unbiased results. By verifying this condition, we can ensure that our one-sample z-test is valid and provides accurate conclusions.

References

• Agresti, A. (2018). Statistical Methods for the Social Sciences. Pearson Education.
• Moore, D. S., & McCabe, G. P. (2017). Introduction to the Practice of Statistics. W.H. Freeman and Company.
• Rosner, B. (2015). Fundamentals of Biostatistics. Cengage Learning.

Further Reading

• One-Sample Z-Test for a Population Proportion: This article provides a detailed explanation of the one-sample z-test for a population proportion, including the assumptions and conditions required for the test.
• Independence in Statistics: This article discusses the concept of independence in statistics, including its importance and how to verify independence in different statistical tests.
• Sample Size Determination: This article provides a detailed explanation of how to determine the sample size required for a one-sample z-test for a population proportion.
  A One-Sample Z-Test for a Population Proportion: Q&A =====================================================

Introduction

In our previous article, we discussed the check for independence in a one-sample z-test for a population proportion. In this article, we will answer some frequently asked questions (FAQs) related to the one-sample z-test for a population proportion.

Q: What is the purpose of a one-sample z-test for a population proportion?

A: The purpose of a one-sample z-test for a population proportion is to determine whether a sample proportion is significantly different from a known population proportion.

Q: What are the assumptions required for a one-sample z-test for a population proportion?

A: The assumptions required for a one-sample z-test for a population proportion are:

• The sample is selected randomly and independently from the population.
• The sample size is sufficiently large (i.e., $n \geq 30$).
• The population size is finite.
• The population proportion is known.

Q: What is the formula for calculating the expected number of successes in a sample?

A: The formula for calculating the expected number of successes in a sample is:

$$m_{0} = n p_{0}$$

where $m_{0}$ is the expected number of successes, $n$ is the sample size, and $p_{0}$ is the known population proportion.

Q: What is the formula for calculating the expected number of failures in a sample?

A: The formula for calculating the expected number of failures in a sample is:

$$n \left( 1 - p_{0} \right)$$

where $n$ is the sample size and $p_{0}$ is the known population proportion.

Q: How do I determine the sample size required for a one-sample z-test for a population proportion?

A: To determine the sample size required for a one-sample z-test for a population proportion, you can use the following formula:

$$n = \frac{\left( Z_{\alpha/2} \right)^{2} p_{0} \left( 1 - p_{0} \right)}{E^{2}}$$

where $n$ is the sample size, $Z_{\alpha/2}$ is the critical value from the standard normal distribution, $p_{0}$ is the known population proportion, and $E$ is the desired margin of error.

Q: What is the difference between a one-sample z-test for a population proportion and a two-sample z-test for a population proportion?

A: The main difference between a one-sample z-test for a population proportion and a two-sample z-test for a population proportion is that a one-sample z-test is used to compare a sample proportion to a known population proportion, while a two-sample z-test is used to compare two sample proportions to each other.

Q: Can I use a one-sample z-test for a population proportion if the sample size is small?

A: No, you should not use a one-sample z-test for a population proportion if the sample size is small. The one-sample z-test requires a sufficiently large sample size (i.e., $n \geq 30$) to be valid.

Q: How do I interpret the results of a one-sample z-test for a population proportion?

A: To interpret the results of a one-sample z-test for a population proportion, you need to compare the calculated z-statistic to the critical value from the standard normal distribution. If the calculated z-statistic is greater than the critical value, you can reject the null hypothesis and conclude that the sample proportion is significantly different from the known population proportion.

Conclusion

In conclusion, the one-sample z-test for a population proportion is a widely used statistical test for determining whether a sample proportion is significantly different from a known population proportion. By understanding the assumptions, formulas, and interpretations of the one-sample z-test, you can use this test to make informed decisions in various fields, including medicine, social sciences, and business.

References

• Agresti, A. (2018). Statistical Methods for the Social Sciences. Pearson Education.
• Moore, D. S., & McCabe, G. P. (2017). Introduction to the Practice of Statistics. W.H. Freeman and Company.
• Rosner, B. (2015). Fundamentals of Biostatistics. Cengage Learning.

Further Reading

• One-Sample Z-Test for a Population Proportion: This article provides a detailed explanation of the one-sample z-test for a population proportion, including the assumptions and conditions required for the test.
• Independence in Statistics: This article discusses the concept of independence in statistics, including its importance and how to verify independence in different statistical tests.
• Sample Size Determination: This article provides a detailed explanation of how to determine the sample size required for a one-sample z-test for a population proportion.