A Random Sample Of 240 Doctors Revealed That 108 Are Satisfied With The Current State Of U.S. Health Care. The Conditions For Inference Are Met. Using $z^*=1.645$, Which Expression Gives A $90\%$ Confidence Interval For The True

Introduction

In the field of statistics, confidence intervals are used to estimate the true value of a population parameter. Given a random sample of 240 doctors, with 108 of them being satisfied with the current state of U.S. health care, we aim to construct a 90% confidence interval for the true proportion of doctors who are satisfied. In this article, we will explore the conditions for inference, the formula for the confidence interval, and the significance of the given $z^*$ value.

Conditions for Inference

Before constructing the confidence interval, we need to ensure that the conditions for inference are met. These conditions include:

• Random sampling: The sample of 240 doctors must be randomly selected from the population of all doctors.
• Independence: The satisfaction of each doctor must be independent of the others.
• Normality: The sampling distribution of the sample proportion must be approximately normal.
• Large sample size: The sample size must be sufficiently large to ensure that the sampling distribution is approximately normal.

In this case, we are given that the conditions for inference are met, which means we can proceed with constructing the confidence interval.

Formula for the Confidence Interval

The formula for the confidence interval is given by:

$$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

where:

• $\hat{p}$ is the sample proportion (108/240 = 0.45)
• $z^*$ is the critical value from the standard normal distribution (1.645)
• $n$ is the sample size (240)

Significance of the Given $z^*$ Value

The given $z^*$ value of 1.645 corresponds to a 90% confidence level. This means that if we were to repeat the sampling process many times, we would expect the confidence interval to contain the true population proportion 90% of the time.

Calculating the Confidence Interval

Now that we have the formula and the values, we can calculate the confidence interval:

$$0.45 \pm 1.645 \sqrt{\frac{0.45(1-0.45)}{240}}$$

$$0.45 \pm 1.645 \sqrt{\frac{0.45 \times 0.55}{240}}$$

$$0.45 \pm 1.645 \sqrt{\frac{0.2475}{240}}$$

$$0.45 \pm 1.645 \sqrt{0.00103125}$$

$$0.45 \pm 1.645 \times 0.0321$$

$$0.45 \pm 0.0527$$

Therefore, the 90% confidence interval for the true proportion of doctors who are satisfied with the current state of U.S. health care is:

$$0.45 - 0.0527 = 0.3973$$

$$0.45 + 0.0527 = 0.5027$$

Conclusion

In conclusion, we have constructed a 90% confidence interval for the true proportion of doctors who are satisfied with the current state of U.S. health care. The interval is given by (0.3973, 0.5027), which means that we are 90% confident that the true proportion of satisfied doctors lies between 39.73% and 50.27%. This interval provides a range of values within which the true population proportion is likely to lie.

Implications

The implications of this confidence interval are significant. If the true proportion of satisfied doctors lies within this interval, it suggests that the current state of U.S. health care is acceptable to a majority of doctors. However, if the true proportion lies outside this interval, it may indicate that there are significant issues with the current system that need to be addressed.

Future Research

Future research could focus on exploring the factors that contribute to doctor satisfaction with the current state of U.S. health care. This could involve analyzing the relationship between doctor satisfaction and various demographic and socioeconomic factors, such as age, gender, and income level. Additionally, researchers could investigate the impact of policy changes on doctor satisfaction and the overall quality of care.

Limitations

One limitation of this study is that it is based on a random sample of 240 doctors, which may not be representative of the entire population of doctors. Additionally, the study only examines the satisfaction of doctors with the current state of U.S. health care, and does not consider other factors that may influence their satisfaction, such as patient outcomes and work-life balance.

Conclusion

Introduction

In our previous article, we explored the construction of a 90% confidence interval for the true proportion of doctors who are satisfied with the current state of U.S. health care. In this article, we will address some of the most frequently asked questions (FAQs) related to this topic.

Q: What is the purpose of a confidence interval?

A: A confidence interval is a statistical tool used to estimate the true value of a population parameter. In this case, we used a confidence interval to estimate the true proportion of doctors who are satisfied with the current state of U.S. health care.

Q: What is the difference between a sample proportion and a population proportion?

A: A sample proportion is the proportion of a specific characteristic (in this case, satisfaction with the current state of U.S. health care) in a sample of data. A population proportion is the true proportion of the characteristic in the entire population.

Q: Why is it important to use a 90% confidence level?

A: A 90% confidence level means that if we were to repeat the sampling process many times, we would expect the confidence interval to contain the true population proportion 90% of the time. This provides a high degree of confidence in the estimate.

Q: What is the significance of the given $z^*$ value?

A: The given $z^*$ value of 1.645 corresponds to a 90% confidence level. This means that if we were to repeat the sampling process many times, we would expect the confidence interval to contain the true population proportion 90% of the time.

Q: Can I use a different confidence level, such as 95% or 99%?

A: Yes, you can use a different confidence level. However, the $z^*$ value will change accordingly. For example, a 95% confidence level would require a $z^*$ value of 1.96, while a 99% confidence level would require a $z^*$ value of 2.58.

Q: How do I interpret the confidence interval?

A: The confidence interval provides a range of values within which the true population proportion is likely to lie. In this case, the 90% confidence interval is (0.3973, 0.5027), which means that we are 90% confident that the true proportion of satisfied doctors lies between 39.73% and 50.27%.

Q: What are some potential limitations of this study?

A: One limitation of this study is that it is based on a random sample of 240 doctors, which may not be representative of the entire population of doctors. Additionally, the study only examines the satisfaction of doctors with the current state of U.S. health care, and does not consider other factors that may influence their satisfaction, such as patient outcomes and work-life balance.

Q: What are some potential applications of this study?

A: This study has several potential applications, including:

• Policy development: The results of this study can inform policy decisions related to the current state of U.S. health care.
• Quality improvement: The study's findings can be used to identify areas for improvement in the current system.
• Research: The study's methodology and results can be used as a starting point for future research on doctor satisfaction and the current state of U.S. health care.

Conclusion

In conclusion, we have addressed some of the most frequently asked questions related to the construction of a 90% confidence interval for the true proportion of doctors who are satisfied with the current state of U.S. health care. We hope that this Q&A article has provided valuable insights and information for readers.