In-Depth Notes on Sample Means and Estimation of Population Means

Introduction

  • The sample mean serves as an estimate for the population mean (µ).
  • A critical question is how accurate the sample mean is and how it reflects plausible population mean values.
  • This document outlines the behavior of the sample mean and how to create plausible intervals for the population mean.

Sample Mean

  • Example: Medicine Tablets
    • Proper weight for tablets: 12 mg.
    • Random sample of 10 tablets has weights: [11.7, 11.8, 12.5, 11.6, 12.1, 11.7, 11.8, 11.4, 11.7, 11.7].
    • The sample mean for this sample is calculated as: ( ar{x} = 11.8 ) mg.
  • The actual interest lies in estimating the population mean µ, which is unknown.

Distribution of Sample Mean

  • Each sample value ( X_i ) is i.i.d. (independently and identically distributed).
  • The sample mean can be treated as a random variable that follows a probability distribution due to its dependence on sample variability.

Central Limit Theorem

  • Distribution of sample means approaches normality under certain conditions:
    1. If the population is normally distributed, then the sample mean is also normally distributed regardless of sample size.
    2. For large sample sizes (n ≥ 30), the sample mean approaches normality even if the population is not normally distributed.
  • The mean of the distribution of sample means is equal to the population mean (µ), and its variance is given by:
    • ( \sigma_{x} = \frac{\sigma}{\sqrt{n}} )

Standard Error of Sample Mean

  • The standard error is an estimate of the variability around the population mean when population standard deviation (σ) is unknown.
  • The (t-distribution) statistical approach is used for samples of all sizes, characterized by degrees of freedom (df = n - 1) when the sample size is small.

Using t-Distribution

  • When n is large, the t-distribution approximates a normal distribution with increased accuracy as degree of freedom increases.
  • The t-values can be referenced from a statistical table to draw probabilistic conclusions.

Confidence Intervals

  • Constructing confidence intervals involves calculating ranges for plausible values of the population mean based on the sample mean:
    • General formula: [ \bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}} ]
    • An example yields a 95% confidence interval of (11.58, 12.02) for µ, indicating that a mean weight of 12 mg is plausible.

Estimator Quality

  • Conditions under which the sample mean is a good estimator of the population mean:
    • Unbiased: E(( \bar{X} )) = µ.
    • Consistent: As the sample size increases, ( \bar{X} ) converges to µ with reduced variance.
    • Efficient: Among unbiased estimators, sample mean has the least variance when the population follows a normal distribution.