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How 95% Confidence Interval for the True Mean formula was proposed The invention of the 95% confidence interval for the true mean is rooted in the development of statistical theory, particularly in the early 20th century. This concept was primarily developed by statisticians seeking ways to make inferences about population parameters based on sample data. Two key figures in this development were Ronald A. Fisher and Jerzy Neyman. Ronald A. Fisher (1890–1962): Fisher introduced the concept of the variance and standard deviation, which are crucial for understanding the distribution of sample means. He contributed significantly to the field of statistics, including the development of the analysis of variance (ANOVA) and the maximum likelihood estimation. While Fisher didn't directly invent the confidence interval as we know it today, his work laid the groundwork by establishing the importance of estimating population parameters from sample statistics and by developing the necessary statistical methods. Jerzy Neyman (1894–1981): Neyman further developed the concept of confidence intervals in the 1930s. He built upon Fisher's work and formalized the idea of an interval estimate, which provides a range of plausible values for an unknown parameter. Neyman's formulation of confidence intervals was part of his broader work on hypothesis testing and estimation, which together with Fisher's contributions, form the basis of modern statistical inference. The specific formula for the 95% confidence interval for the true mean, especially for normally distributed data, is derived from the properties of the normal distribution. Z is approximately , which corresponds to the percentile of the standard normal distribution), and SEM is the standard error of the mean. This formula reflects the idea that if you were to take many samples from a population and calculate the 95% confidence interval for each sample, about 95% of these intervals would contain the true population mean. The development of this formula and the concept of confidence intervals was a significant milestone in statistical theory, allowing researchers to make probabilistic statements about population parameters based on sample data. Problem: A study is conducted concerning the blood pressure of 60 year old women with glaucoma. In the study 200 60-year old women with glaucoma are randomly selected and the sample mean systolic blood pressure is 140 mm Hg and the sample standard deviation is 25 mm Hg. A) Calculate a 95% confidence interval for the true mean systolic blood pressure among the population of 60 year old women with glaucoma. B) Suppose the study above was based on 100 women instead of 200 but the sample mean (140) and standard deviation (25) are the same. Recalculate the 95% confidence interval. Does the interval get wider or narrower? Why?
