Confidence Intervals Notes Please Respond To One Of The Foll

Confidence Intervals Noteplease Respond To One 1 Of The Followin

Confidence Intervals Noteplease Respond To One 1 Of The Followin

" Confidence Intervals " ( Note: Please respond to one [1] of the following two [2] bulleted items) The vast majority of the world uses a 95% confidence in building confidence intervals. Give your opinion on why 95% confidence is so commonplace. Justify your response. Construct a hypothetical 95% confidence interval for a hypothetical case of your choosing. Use your own unique choice of mean, standard deviation, and sample size to calculate the confidence interval.

Select one (1) option provided below and analyze what will happen to your confidence interval based on the option you selected: The confidence changes to 90%. The confidence changes to 99%. The sample size is cut in half. The sample size is doubled. The sample size is tripled. Provide a rationale for your response.

Paper For Above instruction

Confidence intervals are a fundamental concept in statistical analysis, providing a range of values within which a population parameter, such as the mean, is estimated to lie with a specified level of confidence. The widespread use of a 95% confidence level in constructing these intervals stems from a balance between statistical certainty and practical feasibility. This level has become a standard due to its tradition, the law of large numbers, and its acceptability in both scientific research and policy-making. In this paper, I will explore why 95% confidence intervals are so prevalent, construct a hypothetical example to illustrate this concept, and analyze how changes in confidence level and sample size influence the interval's properties.

Why 95% Confidence Is Commonplace

The preference for a 95% confidence level is rooted in historical and practical reasons. Historically, the introduction of confidence intervals by Jerzy Neyman in the early 20th century established 95% as a conventional confidence level, partly due to the mathematical convenience and interpretability it offers. Practically, a 95% confidence level strikes a balance between Type I and Type II errors, providing a degree of certainty that is sufficient for most scientific and social research applications without requiring excessively large sample sizes or overly broad intervals. Additionally, the 95% confidence interval is generally accepted because it aligns with the conventional alpha level of 0.05, meaning there's only a 5% chance of the true parameter falling outside the interval, which is deemed acceptable in many contexts (Cohen, 1988). Overall, this standardization facilitates communication, comparability, and consistency across studies.

Constructing a Hypothetical 95% Confidence Interval

Suppose we are interested in estimating the average height of adult males in a city. Based on a sample of 50 individuals, the sample mean height is 175 cm, with a standard deviation of 10 cm. Using these values, the 95% confidence interval for the population mean can be calculated as follows:

First, determine the standard error (SE):

SE = s / √n = 10 / √50 ≈ 1.414 cm.

Next, identify the critical t-value for 49 degrees of freedom at 95% confidence level. Using a t-distribution table, t ≈ 2.009.

Finally, calculate the confidence interval (CI):

CI = mean ± t SE = 175 ± 2.009 1.414 ≈ 175 ± 2.843

Thus, the 95% confidence interval is approximately (172.157 cm, 177.843 cm). This means we are 95% confident that the true average height of adult males in this city lies within this interval.

Impacts of Changing Confidence Level and Sample Size

Let's analyze the impact on the confidence interval when the parameters change, specifically, when the confidence level shifts from 95% to 90%, or when the sample size is altered.

Changing to 90% Confidence

Reducing the confidence level to 90% means there is a lower degree of certainty about capturing the true population parameter within the interval. The critical t-value decreases from approximately 2.009 to about 1.684 for 49 degrees of freedom, leading to a narrower confidence interval. Consequently, although the interval becomes more precise, there's a higher chance (10% instead of 5%) that the true parameter falls outside this range. This trade-off often makes 90% confidence intervals preferable when precision is prioritized over certainty (Cumming & Finch, 2005).

Doubling the Sample Size

Increasing the sample size impacts the confidence interval significantly. When the sample size is doubled, n becomes 100, and the standard error reduces to:

SE = 10 / √100 = 10 / 10 = 1 cm.

Using the same t-value for 99 degrees of freedom at 95% confidence (approximately 1.984), the confidence interval becomes:

175 ± 1.984 * 1 ≈ 175 ± 1.984, which is (173.016 cm, 176.984 cm).

The wider dataset reduces the standard error, resulting in a narrower interval that provides a more precise estimate of the population mean. This exemplifies the benefit of larger samples in statistical estimation, decreasing uncertainty and yielding more reliable confidence intervals (Schmidt & Hunter, 2015).

Conclusion

In summary, the choice of confidence level and sample size directly influences the width and reliability of confidence intervals. A 95% confidence interval is a balanced standard widely accepted for its practicality and interpretability. Adjustments to confidence level and sample size can either narrow or widen the interval and affect the degree of certainty associated with the estimate. Larger samples tend to produce more precise intervals, whereas higher confidence levels increase the interval's width but reduce the risk of missing the true parameter. Understanding these relationships is vital for effective statistical analysis and communication of results.

References

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