2024-10-31

Introduction

  • Point estimation is a common statistical model used to make inferences on a population’s unknown parameter by using a sample data.
  • Unlike interval estimation, point estimation goes through a process to calculate only a single estimate for the parameter of interest.
  • The parameter can be any statistical value (e.g. population mean, variance, standard deviation, etc.)

Methods to Calculate Point Estimators

Including, but not limited to:

  • Methods of moments (MoMs)
  • Maximum likelihood estimation (MLE)
  • Bayesian Method
  • These methods are dependent on the nature of the parameter of interest and what information our sample data has already provided for us.

Common Point Estimators

  1. Sample mean is the point estimator of the population mean: \[ \overline{x} = (\Sigma x_i) / n \]

  2. Sample variance is the point estimator of the population variance: \[ s^2 = \Sigma(x_i - \overline{x})^2/(n-1) \]

  3. Sample standard deviation is the point estimator of the population standard deviation: \[ S = \sqrt{\Sigma(x-\overline{x})^2/(n-1)} \]

  • \(x_i\) represents a single observation
  • \(n\) represents the sample size

Example: Sampling Distribution of Sample Means

  • Using data set tips, we will first manipulate the columns total_bill and tip to calculate the tip percentage for each observation, denoted as tip_percentage:
data(tips)
tips = subset(tips, select = -c(sex, smoker, day, time, size))
tips$tip_percentage <- round((tips$tip / tips$total_bill) * 100, 2)
summary(tips)
##    total_bill         tip         tip_percentage 
##  Min.   : 3.07   Min.   : 1.000   Min.   : 3.56  
##  1st Qu.:13.35   1st Qu.: 2.000   1st Qu.:12.91  
##  Median :17.80   Median : 2.900   Median :15.47  
##  Mean   :19.79   Mean   : 2.998   Mean   :16.08  
##  3rd Qu.:24.13   3rd Qu.: 3.562   3rd Qu.:19.15  
##  Max.   :50.81   Max.   :10.000   Max.   :71.03
  • In this instance, we will build random samples of the tip percentages with the formula: \(\overline{x} = (x_1+x_2+x_3+...+x_{12})/12\)

Dot Plot Sampling Dist.

  • Note that this is not the most optimal graph to analyze the sampling distribution because dot plots are more focused on each observation.

Histogram Sampling Dist.

  • The histogram is the most optimal way to see where we can estimate the population mean as depicted by the density line.

2D Density Plot of Sampling Dist.