STA291 Statistical Methods Lecture 16

Lecture 15 Review Assume that a school district has 10,000 6th graders. In this district, the average weight of a 6th grader is 80 pounds, with a standard deviation of 20 pounds. Suppose you draw a random sample of 50 students. A) What is the probability that the average weight will be less than 75 pounds?

So far … We know a little bit about: o Collecting data, and on what scale to do it o How to describe it, graphically and numerically o What to describe about it: o center and spread, if quantitative o proportion in each category, if qualitative o Probability, including: o basic rules o random variables, including discrete (binomial) and continuous (normal) examples o Sampling distributions 3

Central Limit Theorem Thanks to the CLT … We know is approximately standard normal (for sufficiently large n, even if the original distribution is discrete, or skewed). Ditto 4

Two primary types of statistical inference o Estimation o using information from the sample (statistic’s value, for example) to make an informed (mathematically justifiable) guess about a characteristic of the population o Hypothesis, or SignificanceTesting o using information from the sample to make an informed decision about some aspect of the population 5

Statistical Inference: Estimation o Inferential statistical methods provide predictions about characteristics of a population, based on information in a sample from that population o For quantitative variables, we usually estimate the population mean (for example, mean household income) o For qualitative variables, we usually estimate population proportions (for example, proportion of people voting for candidate A) 6

Two Types of Estimators o Point Estimate o A single number that is the best guess for the parameter o For example, the sample mean is usually a good guess for the population mean o Interval Estimate o A range of numbers around the point estimate o To give an idea about the precision of the estimator o For example, “the proportion of people voting for A is between 67% and 73%” 7

A Good Estimator is … o unbiased: Centered around the true parameter o consistent: Gets closer to the true parameter as the sample size gets larger o efficient: Has a standard error that is as small as possible 8

Unbiased o Already have two examples of unbiased estimators … o Expected Value of the ’s:  —that makes an unbiased estimator of . o Expected Value of the ’s: p—that makes an unbiased estimator of p. o Third example: 9

Efficiency 10 o An estimator is efficient if its standard error is small compared to other estimators o Such an estimator has high precision o A good estimator has small standard error and small bias (or no bias at all)

Bias versus Efficiency 11 AB CD

Looking back o Recap of descriptive statistics, probability work o Inferential statistics: o estimation o hypothesis testing o Considerations in estimation o bias o consistency o efficiency