STA291Statistical Methods

Lecture 16

Lecture 15 ReviewAssume 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:oCollecting data, and on what scale to do

itoHow to describe it, graphically and

numericallyoWhat to describe about it:ocenter and spread, if quantitativeoproportion in each category, if qualitative

oProbability, including:obasic ruleso random variables, including discrete

(binomial) and continuous (normal) examplesoSampling 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

n

X

npp

pp

1

ˆ

4

Two primary types of statistical inference

o Estimationousing 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 SignificanceTestingousing information from the sample to make an informed decision about some aspect of the population

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Statistical Inference: Estimationo 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)

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Two Types of Estimatorso Point EstimateoA single number that is the best guess for the

parametero For example, the sample mean is usually a

good guess for the population mean

o Interval EstimateoA range of numbers around the point

estimateoTo give an idea about the precision of the

estimatoro For example, “the proportion of people voting

for A is between 67% and 73%”7

A Good Estimator is …ounbiased: Centered around the true

parameter

oconsistent: Gets closer to the true parameter as the sample size gets larger

oefficient: Has a standard error that is as small as possible

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Unbiasedo Already have two examples of

unbiased estimators …o Expected Value of the ’s: m—

that makes an unbiased estimator of m.

o Expected Value of the ’s: p—that makes an unbiased estimator of p.

o Third example: 9

X X

p̂ p̂

22

1

1xx

ns i

Efficiency

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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

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A B

C D

Looking back

oRecap of descriptive statistics, probability workoInferential statistics:o estimationo hypothesis testingoConsiderations in estimationo biaso consistencyo efficiency