A Random Sample Of 240 Doctors Revealed That 108 Are Satisfied With The Current State Of US Health Care. The Conditions For Inference Are Met. Using $Z^{\star}=1.645$, Which Expression Gives A $90 \%$ Confidence Interval For The

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Introduction

In the field of statistics, constructing a confidence interval is a crucial aspect of making inferences about a population based on a sample of data. Given a random sample of 240 doctors, with 108 of them being satisfied with the current state of US health care, we aim to determine the expression that provides a 90% confidence interval for the population proportion. This article will guide you through the process of constructing a confidence interval using the given information and the specified conditions for inference.

Conditions for Inference

Before we proceed with constructing the confidence interval, it is essential to verify that the conditions for inference are met. These conditions include:

  • Random sampling: The sample of 240 doctors is randomly selected from the population of interest.
  • Independence: The observations in the sample are independent of each other.
  • Normality: The sampling distribution of the sample proportion is approximately normal.
  • Fixed sample size: The sample size is fixed, and we are not dealing with a changing sample size.

In this scenario, we are given that the conditions for inference are met, which allows us to proceed with constructing the confidence interval.

Constructing the Confidence Interval

To construct a 90% confidence interval for the population proportion, we use the following formula:

p^±Z⋆p^(1−p^)n\hat{p} \pm Z^{\star} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where:

  • p^\hat{p} is the sample proportion (108/240 = 0.45)
  • Z⋆Z^{\star} is the critical value from the standard normal distribution for a 90% confidence interval (1.645)
  • nn is the sample size (240)

Substituting the values into the formula, we get:

0.45±1.6450.45(1−0.45)2400.45 \pm 1.645 \sqrt{\frac{0.45(1-0.45)}{240}}

Calculating the Margin of Error

To calculate the margin of error, we need to evaluate the expression under the square root:

0.45(1−0.45)240\sqrt{\frac{0.45(1-0.45)}{240}}

0.45×0.55240\sqrt{\frac{0.45 \times 0.55}{240}}

0.2475240\sqrt{\frac{0.2475}{240}}

0.00103125\sqrt{0.00103125}

0.03210.0321

Constructing the Confidence Interval

Now that we have the margin of error, we can construct the 90% confidence interval:

0.45±1.645×0.03210.45 \pm 1.645 \times 0.0321

0.45±0.05280.45 \pm 0.0528

The 90% confidence interval for the population proportion is:

0.3972≤p≤0.50280.3972 \leq p \leq 0.5028

Conclusion

In this article, we have constructed a 90% confidence interval for the population proportion of doctors satisfied with the current state of US health care. Using the given sample proportion, sample size, and critical value from the standard normal distribution, we have calculated the margin of error and constructed the confidence interval. The resulting interval provides a range of values within which the true population proportion is likely to lie with 90% confidence.

Discussion

The confidence interval provides a range of values within which the true population proportion is likely to lie. This interval can be used to make inferences about the population proportion, such as determining whether the true population proportion is greater than or less than a certain value. Additionally, the confidence interval can be used to estimate the precision of the sample proportion.

Limitations

One limitation of this study is that the sample size is relatively small (n = 240). A larger sample size would provide a more precise estimate of the population proportion. Additionally, the study assumes that the conditions for inference are met, which may not be the case in all scenarios.

Future Research

Future research could involve collecting a larger sample size to improve the precision of the estimate. Additionally, researchers could investigate the factors that contribute to a doctor's satisfaction with the current state of US health care.

References

  • [1] Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. (2010). Elementary statistics: A step-by-step approach. McGraw-Hill.
  • [3] Agresti, A., & Franklin, C. A. (2013). Statistics: The art and science of learning from data. Pearson Education.

Introduction

In our previous article, we constructed a 90% confidence interval for the population proportion of doctors satisfied with the current state of US health care. In this article, we will address some of the frequently asked questions (FAQs) related to the study.

Q: What is the purpose of constructing a confidence interval?

A: The purpose of constructing a confidence interval is to provide a range of values within which the true population parameter is likely to lie. In this case, we constructed a 90% confidence interval for the population proportion of doctors satisfied with the current state of US health care.

Q: What is the difference between a point estimate and a confidence interval?

A: A point estimate is a single value that estimates the population parameter, while a confidence interval provides a range of values within which the true population parameter is likely to lie.

Q: How is the margin of error calculated?

A: The margin of error is calculated using the formula:

p^(1−p^)n\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where:

  • p^\hat{p} is the sample proportion
  • nn is the sample size

Q: What is the critical value from the standard normal distribution for a 90% confidence interval?

A: The critical value from the standard normal distribution for a 90% confidence interval is 1.645.

Q: How is the confidence interval constructed?

A: The confidence interval is constructed using the formula:

p^±Z⋆p^(1−p^)n\hat{p} \pm Z^{\star} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where:

  • p^\hat{p} is the sample proportion
  • Z⋆Z^{\star} is the critical value from the standard normal distribution
  • nn is the sample size

Q: What is the 90% confidence interval for the population proportion of doctors satisfied with the current state of US health care?

A: The 90% confidence interval for the population proportion of doctors satisfied with the current state of US health care is:

0.3972≤p≤0.50280.3972 \leq p \leq 0.5028

Q: What are the limitations of this study?

A: One limitation of this study is that the sample size is relatively small (n = 240). A larger sample size would provide a more precise estimate of the population proportion. Additionally, the study assumes that the conditions for inference are met, which may not be the case in all scenarios.

Q: What are some potential future research directions?

A: Some potential future research directions include:

  • Collecting a larger sample size to improve the precision of the estimate
  • Investigating the factors that contribute to a doctor's satisfaction with the current state of US health care

Q: What are some real-world applications of confidence intervals?

A: Confidence intervals have numerous real-world applications, including:

  • Estimating the population proportion of customers who are satisfied with a product or service
  • Determining the effectiveness of a new treatment or intervention
  • Estimating the population mean of a continuous variable

Conclusion

In this article, we have addressed some of the frequently asked questions (FAQs) related to the study. We hope that this Q&A article has provided a better understanding of the study and its results.

Discussion

Confidence intervals are a powerful tool for making inferences about a population based on a sample of data. By understanding how to construct and interpret confidence intervals, researchers and practitioners can make more informed decisions and improve the accuracy of their estimates.

Limitations

One limitation of this study is that the sample size is relatively small (n = 240). A larger sample size would provide a more precise estimate of the population proportion. Additionally, the study assumes that the conditions for inference are met, which may not be the case in all scenarios.

Future Research

Future research could involve collecting a larger sample size to improve the precision of the estimate. Additionally, researchers could investigate the factors that contribute to a doctor's satisfaction with the current state of US health care.

References

  • [1] Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. (2010). Elementary statistics: A step-by-step approach. McGraw-Hill.
  • [3] Agresti, A., & Franklin, C. A. (2013). Statistics: The art and science of learning from data. Pearson Education.