A Newsletter Publisher Believes That Less Than 29% Of Their Readers Own A Rolls Royce. Is There Sufficient Evidence At The 0.10 Level To Substantiate The Publisher's Claim?State The Null And Alternative Hypotheses For The Above Scenario.- \[$ H_0

Introduction

In this article, we will delve into a statistical scenario where a newsletter publisher makes a claim about the ownership of Rolls Royce vehicles among their readers. The publisher believes that less than 29% of their readers own a Rolls Royce. We will examine whether there is sufficient evidence to support this claim at a 0.10 significance level.

Null and Alternative Hypotheses

To begin our analysis, we need to define the null and alternative hypotheses.

• Null Hypothesis (H0): The proportion of readers who own a Rolls Royce is greater than or equal to 29% (p ≥ 0.29).
• Alternative Hypothesis (H1): The proportion of readers who own a Rolls Royce is less than 29% (p < 0.29).

Statistical Significance

The significance level, denoted by α, is set at 0.10. This means that if the p-value is less than 0.10, we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.

Sample Data

Let's assume that the newsletter publisher conducts a survey of 1000 readers and finds that 270 of them own a Rolls Royce. We can use this sample data to calculate the sample proportion (p̂) and the standard error (SE).

• Sample Proportion (p̂): p̂ = (number of readers who own a Rolls Royce) / (total number of readers) = 270 / 1000 = 0.27
• Standard Error (SE): SE = √(p̂(1-p̂)/n) = √(0.27(1-0.27)/1000) = 0.015

Hypothesis Testing

We can now perform a hypothesis test using the sample proportion and the standard error.

• Test Statistic: z = (p̂ - p) / SE = (0.27 - 0.29) / 0.015 = -1.33
• p-value: p-value = P(Z < -1.33) = 0.091

Conclusion

Based on the p-value of 0.091, which is greater than the significance level of 0.10, we fail to reject the null hypothesis. This means that there is insufficient evidence to support the publisher's claim that less than 29% of their readers own a Rolls Royce.

Discussion

In this scenario, the null hypothesis states that the proportion of readers who own a Rolls Royce is greater than or equal to 29%. The alternative hypothesis states that the proportion of readers who own a Rolls Royce is less than 29%. The sample data shows that 270 out of 1000 readers own a Rolls Royce, which gives a sample proportion of 0.27. The standard error is calculated as 0.015.

The hypothesis test is performed using the sample proportion and the standard error. The test statistic is calculated as -1.33, and the p-value is found to be 0.091. Since the p-value is greater than the significance level of 0.10, we fail to reject the null hypothesis.

Implications

The results of this analysis have implications for the newsletter publisher. If the publisher wants to make a claim about the ownership of Rolls Royce vehicles among their readers, they need to collect more data or use a more sensitive test to detect a difference in proportions.

Limitations

This analysis has some limitations. The sample size is relatively small, and the survey may not be representative of the entire population of readers. Additionally, the survey may be subject to biases or errors that can affect the accuracy of the results.

Future Research

Future research could involve collecting more data or using a more sensitive test to detect a difference in proportions. Additionally, the survey could be designed to collect more information about the readers, such as their demographics or interests, to better understand the relationship between Rolls Royce ownership and other factors.

Conclusion

Introduction

In our previous article, we analyzed a statistical scenario where a newsletter publisher makes a claim about the ownership of Rolls Royce vehicles among their readers. We examined whether there is sufficient evidence to support this claim at a 0.10 significance level. In this article, we will answer some frequently asked questions (FAQs) related to this scenario.

Q: What is the null and alternative hypothesis in this scenario?

A: The null hypothesis (H0) states that the proportion of readers who own a Rolls Royce is greater than or equal to 29% (p ≥ 0.29). The alternative hypothesis (H1) states that the proportion of readers who own a Rolls Royce is less than 29% (p < 0.29).

Q: What is the significance level in this scenario?

A: The significance level, denoted by α, is set at 0.10. This means that if the p-value is less than 0.10, we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.

Q: What is the sample proportion (p̂) in this scenario?

A: The sample proportion (p̂) is calculated as the number of readers who own a Rolls Royce divided by the total number of readers. In this scenario, p̂ = 270 / 1000 = 0.27.

Q: What is the standard error (SE) in this scenario?

A: The standard error (SE) is calculated as the square root of the product of the sample proportion, one minus the sample proportion, and the sample size. In this scenario, SE = √(0.27(1-0.27)/1000) = 0.015.

Q: What is the test statistic (z) in this scenario?

A: The test statistic (z) is calculated as the difference between the sample proportion and the population proportion, divided by the standard error. In this scenario, z = (0.27 - 0.29) / 0.015 = -1.33.

Q: What is the p-value in this scenario?

A: The p-value is the probability of observing a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. In this scenario, the p-value is 0.091.

Q: What is the conclusion of this analysis?

A: Based on the p-value of 0.091, which is greater than the significance level of 0.10, we fail to reject the null hypothesis. This means that there is insufficient evidence to support the publisher's claim that less than 29% of their readers own a Rolls Royce.

Q: What are the implications of this analysis?

A: The results of this analysis have implications for the newsletter publisher. If the publisher wants to make a claim about the ownership of Rolls Royce vehicles among their readers, they need to collect more data or use a more sensitive test to detect a difference in proportions.

Q: What are the limitations of this analysis?

A: This analysis has some limitations. The sample size is relatively small, and the survey may not be representative of the entire population of readers. Additionally, the survey may be subject to biases or errors that can affect the accuracy of the results.

Q: What are some potential future research directions?

A: Future research could involve collecting more data or using a more sensitive test to detect a difference in proportions. Additionally, the survey could be designed to collect more information about the readers, such as their demographics or interests, to better understand the relationship between Rolls Royce ownership and other factors.

Conclusion

In conclusion, this Q&A article provides a summary of the statistical analysis of the Rolls Royce claim. We hope that this article has been helpful in answering some of the frequently asked questions related to this scenario.