Assignment 1: Confidence Interval For One Sample Continuous

Assignment1confidence Interval For One Sample Continuous Outcome Sam

Suppose you are a junior researcher of a research team. You want to know the true mean age of 2,500 freshmen students of MSU. You have randomly selected 144 freshmen (n=144), and collected their age. The mean age x̄ = 25.3 years with a standard deviation s = 5.7. You are asked to compute a 95% confidence interval (CI) for the true mean age of freshmen and to interpret your findings.

Additionally, assume you have collected data from only 28 students (n=28), with a mean x̄ = 24.2 years and standard deviation s = 4.3. You are to compute a 95% confidence interval for the true mean age, interpret the findings, and discuss the differences between data collected from 144 students versus 28 students, including the reasons for these differences. Lastly, in a different scenario, in a sample of 1,540 patients attending an emergency room, 231 patients were diagnosed with cardiac failure. You are to calculate a 90% confidence interval for the proportion of patients with heart failure and interpret the findings.

Paper For Above instruction

In medical and social science research, understanding the true mean or proportion of a population parameter is crucial for making informed decisions. Confidence intervals (CIs) serve as a fundamental statistical tool that provides a range of plausible values for the population parameter based on sample data. This paper aims to illustrate the computation and interpretation of confidence intervals for both continuous (mean ages) and dichotomous (proportion of patients with cardiac failure) outcomes, using scenarios with varying sample sizes.

Confidence Interval for a Continuous Outcome with a Sample Size ≥30

The first scenario involves a sample size of 144 students, which exceeds the threshold of 30, allowing the use of the t-distribution to estimate the confidence interval for the population mean. Given the sample mean (x̄) is 25.3 years and the sample standard deviation (s) is 5.7, the 95% confidence interval can be calculated as follows:

The standard error of the mean (SEM) is:

SEM = s / √n = 5.7 / √144 = 5.7 / 12 = 0.475

Using the t-distribution with df = n - 1 = 143 degrees of freedom, the critical t-value for a 95% CI is approximately 1.976 (obtained from t-tables or statistical software). The margin of error (ME) is:

ME = t SEM = 1.976 0.475 ≈ 0.938

Thus, the 95% confidence interval is:

25.3 ± 0.938, which ranges from approximately 24.36 to 26.24 years.

This interval suggests that we are 95% confident that the true mean age of all freshmen at MSU lies between 24.36 and 26.24 years.

Interpretation of the CI

The computed confidence interval indicates that, based on the sample data, we can be 95% confident that the average age of the entire freshman population falls within the range of approximately 24.36 to 26.24 years. Such an interval accounts for sampling variability and provides a plausible range for the population parameter.

Confidence Interval with a Smaller Sample Size (

In the second scenario, a sample of 28 students is used to estimate the mean age, with x̄ = 24.2 years and s = 4.3. Because the sample size is less than 30, the t-distribution with df = 27 is appropriate for the calculation.

The SEM is:

SEM = 4.3 / √28 ≈ 4.3 / 5.292 ≈ 0.813

The critical t-value for a 95% CI with 27 degrees of freedom is approximately 2.051. The margin of error is:

ME = 2.051 * 0.813 ≈ 1.67

The confidence interval is:

24.2 ± 1.67, resulting in a range from roughly 22.53 to 25.87 years.

This indicates with 95% confidence that the true mean age of all freshmen is between approximately 22.53 and 25.87 years.

Differences Between Larger and Smaller Samples

Notable differences arise when comparing datasets of differing sizes. Larger samples (n=144) tend to produce narrower confidence intervals, reflecting increased precision and reduced sampling variability. Smaller samples (n=28) usually lead to broader intervals, indicating greater uncertainty about the true parameter estimate. The primary reason for these differences is the standard error's inverse relationship with sample size; as n increases, SEM decreases, resulting in more precise estimates.

Moreover, larger samples more reliably approximate the population distribution, leading to more accurate inferences. Smaller samples are more susceptible to sampling fluctuations, which can inflate the margin of error and widen the confidence interval, sometimes reducing the confidence in the estimate.

Confidence Interval for a Proportion with Dichotomous Data

The third scenario involves estimating the proportion of patients with cardiac failure in a sample of 1,540 patients, where 231 patients were diagnosed. The point estimate of the proportion (p̂) is:

p̂ = 231 / 1540 ≈ 0.15

To calculate the 90% confidence interval for this proportion, we first determine the standard error (SE) for the proportion:

SE = √[p̂ (1 - p̂) / n] = √[0.15 * 0.85 / 1540] ≈ √(0.1275 / 1540) ≈ √0.0000828 ≈ 0.0091

The critical z-value for a 90% confidence level is approximately 1.645. The margin of error (ME) is:

ME = 1.645 * 0.0091 ≈ 0.015

Thus, the confidence interval is:

0.15 ± 0.015, or from approximately 0.135 to 0.165, meaning we are 90% confident that between 13.5% and 16.5% of ER patients have cardiac failure.

Interpretation of the Proportion CI

This interval implies that, based on the sample, it is plausible that the true proportion of ER patients with cardiac failure lies within the range of about 13.5% to 16.5%. The use of the confidence level indicates the long-term reliability of this interval in capturing the true population parameter, considering the sampling process.

Conclusion

Confidence intervals provide vital information about the precision and variability of sample estimates in relation to population parameters. Larger samples tend to produce narrower, more reliable CIs, enhancing the accuracy of inferences. Conversely, smaller samples yield broader intervals, representing increased uncertainty. Calculating and interpreting these intervals accurately allows researchers to make more informed conclusions about the population, guiding clinical and policy decisions effectively. The methods outlined herein highlight the importance of adequate sample sizes and appropriate statistical techniques in scientific research.

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