Confidence Intervals Calculation 5: Calculating Confidence I

Confidence Intervals Calculation 5 Calculating Confidence Intervals Berline

Calculate confidence intervals for the quantitative variables in the Heart Rate Dataset, which includes values taken before and after exercise, categorized by gender (male and female). The data follows a normal distribution, with a known sample size of 150, a sample mean of 15, and a sample standard deviation of 2. The population size is 200, with 108 males and 92 females. Using the data, calculate confidence intervals at the 95% and 99% levels, including the z-scores for these confidence levels, and interpret the results for the population mean.

Additionally, perform a hypothesis test for the difference between two sample means with sample sizes of 40 and 50, means of 50 and 51, and standard deviations of 1.2 and 1.8 respectively. Calculate the z-score for the difference, determine the confidence intervals at the 90%, 95%, and 99% levels, and interpret these intervals in the context of comparing the two population means.

Paper For Above instruction

Confidence intervals are vital statistical tools that provide a range of values within which a population parameter, such as a mean, is likely to lie with a specified level of confidence. In research and data analysis, calculating confidence intervals helps researchers understand the precision of their estimates and the degree of uncertainty associated with sample data. This paper discusses the calculation of confidence intervals using z-scores, specifically focusing on the Heart Rate Dataset, which involves analyzing resting and post-exercise heart rates categorized by gender. Additionally, the paper explores the comparison of two population means through hypothesis testing and confidence interval estimation.

Introduction

Confidence intervals serve as an essential component of inferential statistics, allowing researchers to make probabilistic statements about population parameters based on sample data. They offer a quantifiable measure of uncertainty and are widely used in various fields such as healthcare, economics, and social sciences. This paper explains the methodology of calculating confidence intervals for means using the z-distribution, emphasizing two key applications: estimating a single population mean and comparing two population means.

Calculating Confidence Intervals for a Single Population Mean

Using the Heart Rate Dataset, we aim to determine the confidence intervals for the population mean of heart rates during rest or after exercise. The dataset's sample size (n) is 150, with a sample mean (\(\bar{x}\)) of 15 and a sample standard deviation (s) of 2. The assumption that the population distribution follows a normal curve justifies the use of z-scores.

The standard error (SE) of the mean is calculated as \(SE = \frac{s}{\sqrt{n}}\). Substituting the provided values yields \(SE = \frac{2}{\sqrt{150}} \approx 0.1633\). Because the population standard deviation (\(\sigma\)) is unknown, and the sample size is large, the z-distribution is appropriate for decision-making (Gelman & Hill, 2006).

The z-scores associated with the 95% and 99% confidence levels are 1.96 and 2.576, respectively. The confidence interval formula is: CI = \(\bar{x} \pm z \times SE\).

At the 95% confidence level, the interval is:

15 ± 1.96 × 0.1633 → (15 - 0.319, 15 + 0.319) → (14.68, 15.32)

Similarly, at the 99% confidence level:

15 ± 2.576 × 0.1633 → (15 - 0.422, 15 + 0.422) → (14.58, 15.42)

Interpreting the Results for a Single Mean

The calculated confidence intervals suggest that, with 95% confidence, the true mean heart rate lies between approximately 14.68 and 15.32 beats per minute, and with 99% confidence, it lies between 14.58 and 15.42. These narrow intervals indicate a high level of precision in estimating the population mean from the sample data (Cumming & Finch, 2005).

Comparing Two Population Means

Next, the analysis extends to compare two independent samples of different sizes—40 and 50—measuring some heart rate-related variable. The sample means are 50 and 51, with standard deviations of 1.2 and 1.8, respectively. The difference in sample means is 1.

The standard error of the difference between two means is calculated as:

\(SE_{diff} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)

Plugging in the values:

\(SE_{diff} = \sqrt{\frac{1.2^2}{40} + \frac{1.8^2}{50}} = \sqrt{\frac{1.44}{40} + \frac{3.24}{50}} \approx \sqrt{0.036 + 0.0648} \approx \sqrt{0.1008} \approx 0.317\)

Using the z-distribution, the confidence intervals for the difference are computed with the z-scores for specified confidence levels: 1.645 (90%), 1.96 (95%), and 2.576 (99%).

For example, at 95% confidence:

Difference = 1 ± 1.96 × 0.317 → (1 - 0.621, 1 + 0.621) → (0.379, 1.621)

Interpreting this, we are 95% confident that the true difference in means between the populations lies between approximately 0.38 and 1.62. Since the entire interval is positive, this suggests a statistically significant difference favoring the second group (Moore & McCabe, 2009).

Conclusion

Calculating confidence intervals provides insight into the likely range of a population parameter, such as the mean or the difference between two means. Using the z-score method, we can quantify the uncertainty associated with sample estimates, which is essential in making reliable inferences about populations. The importance of understanding these concepts is especially relevant in health sciences, where such analyses inform clinical decisions and policy-making.

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