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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
5
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)
6
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
8
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
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
A B
C D
Looking back
oRecap of descriptive statistics, probability workoInferential statistics:o estimationo hypothesis testingoConsiderations in estimationo biaso consistencyo efficiency