ECON1203/ECON2292 Business and Economic

Statistics

Week 6

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Week 6 topics

z Normal distribution z Calculating areas under the normal curvez Normal approximation to the Binomial

z Concept of an estimatorz Properties of estimators

z Key referencesz Keller 8.2, 9.2 pp 321-6, 10.1

Normal distribution

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Normal distributionz Plays a pivotal role in statistical theory & practice

z In practice many continuous variables, e.g. heights of men & women, have distributions that are bell-shaped and well-approximated by normal distributions

z Will play a prominent role in upcoming discussion of statistical inference & estimators

z A normal rv X is a continuous rvz P(X = x) = 0 for every xz P(a < X < b) = area under pdf curve between a & bz Completely characterized by its mean P & variance V2

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Normal distribution z Graphically, the probability density function (pdf) is

symmetric, unimodal & bell-shapedz Î mean=median=mode

z Basic features includez Range of “support” is unlimited, so -����[����z Despite unlimited range little area in tails of a normal distribution

z 4.5% outside P ±2V�; 0.3% outside P ±3V��(confirm from tables)

z Relevant tables in Keller,Table 3, p. B8-B9z Also in tutorial program

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Normal distribution

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Normal distribution z If X is normally distributed with mean P and variance V2 then we write X ~ N(P,V2)

z The algebraic formula for pdf of X:

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Normal distribution as a model“All models are wrong but some are useful” George Box

z When a normal distribution is appropriate?z No “exact” way to justify! z Plot a histogram to see if it has a bell shapez Using past experiencez There might be theoretical or context-specific reasons to

justify why a normal is usefulz Can also use statistical hypothesis testing methods

to assess whether a normal “model” might be reasonablez Refer Week 12

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Standard normalz Recall Z scores

z Z = (X-P)/Vz This standardization yields a rv with zero mean & standard

deviation of onez How are Z & X related?

Z = a + bX (where a = -(P /V) and b = 1/V)z So Z is a linear function (linear transformation) of Xz An important theorem tells us the following: Linear

combinations of normal rv’s are also normalz So Z is a normal rv and called a standard normal rv.

Write Z~(0,1)z This result assists calculations of probabilities

z Any probability calculation involving X~N(P,V2) translates into an equivalent statement using Z~N(0,1)

Calculating normal probabilities

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formsz “BES” tables provide P(0 < Z < z), z>0z Keller, P(-��=�]�

z Use the table (next slide)to compute

z What is P(0<Z<0.50)?z What is P(0<Z<1.00)?z What is P(0<Z<1.96)?

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Calculating normal probabilitiesz Suppose time (in minutes) taken

to assemble a computer assumed X~N(50,100). What is probability that a computer is assembled in 45 to 60 minutes?

z Probability of assembly time being between 45 & 60 minutes is 0.5328

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Calculating normal percentiles

z Tables can be used to solve 2 types of problemsz Given a particular z find P(0 < Z < z), orz Given a particular probability A find zA such that

P(Z > zA) = A or P(Z < zA) = 1 - A z Note that zA is the 100(1-A)th percentile of a

standard normal

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Calculating normal percentiles z Use tables to verify that

z0.025 = 1.96z What isz 97.5th percentile?

z 2.5th percentile?

z 97.5th percentile in computer assembly line example?

1.96=(X0.025 – 50)/10

X0.025 =(1.96)10+50=69.6

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Normal approximation to the binomialz Used formula or tables to

evaluate probabilities for a binomial rvz Convenient for small number

of trials ( n )z What if n is large?

z Important application of normal distribution is to approximatethe binomial for large nz See example from Keller, Fig.

9.8, p. 310, where n=20, p=0.5z Do you think approximation

would be better or worse if p=0.2?

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Normal approximation to the binomial z Denote binomial rv by XBz Know that E(XB)=np=10 & Var(XB)= np(1-p)=5z Natural to choose approximating normal rv as

XN~N(10,5)z How good is the approximation?

z Consider P(10�XB ���� ����������������� �����whileP(10�XN ���� P(0�=������ �����

z What went wrong with the approximation?z Would you approximate P(XB=10) by P(XN=10)?

Normal approximation to the binomial ...z Need a continuity correction (Keller p. 311)z Would approximate P(XB=10) by

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Normal approximation to the binomial

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Airline mealsz On today’s flight all 160

passengers offered a lunch choice of beef or chicken

z Past data indicates 60% choose beef over chicken

z Passenger choices appear to be independent

z On this flight what is the probability that more than 110 passengers will choose beef?

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Estimationz Inferential statisticsz Extracting information about population parameters on

basis of sample statisticsz “Past data indicates 60% choose beef over chicken”z What does a sample mean tell us about the population mean?

z In practical situations parameters are unknown because they are difficult or impossible to determinez Using sample statistics may be only practical

alternative

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Estimation z Estimatorsz Consider a generic parameter T� that characterizes

the pdf, f(x), of a rv Xz Suppose X1, X2, , Xn is a sample of size n from f(x)z A statistic is any function of sample dataz An estimator is a statistic whose purpose is to

estimate a parameter or some function thereofz A point estimator is simply a formula (rule) for

combining sample information to produce a single number to estimate T

Estimation ...z Estimators are random variables (because

they are a function of rv’s X1, X2, , Xn )z Examples of point estimatorsz Sample mean is a point estimator for the population

meanz Sample variance is an estimator of the population

variancez Why does it make no sense to expect an estimator to

always produce an estimate equal to the parameter of interest?

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Properties of estimatorsz Sample mean is a ‘natural’ choice of estimator for

the population meanz But there may be other (better?) estimatorsz Why not use (n-1)s2/n as an estimator for V2?

z Desirable properties of estimatorsz Unbiasedness: On average, does the estimator achieve

the correct value?z Consistency: As sample size gets larger does the

probability that the estimator deviates from the parameter by more than a ‘small’ amount get small?

z Relative efficiency: If there are two competing estimators of a parameter, does one have less expected dispersion?

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Properties of estimators

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Properties of estimators

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Progress report #4z Have introduced a selection of distributions z Binomial, uniform & normalz These enable us to model a range of phenomenaz Normal also plays a key role in theory of estimation

z Have introduced the basics of estimationz Need to understand better the notion of a point estimator

as a rvz Leads us to sampling distributionsz Role of Normal distribution in theory of estimation comes

through the Central Limit Theoremz Can also use interval rather then point estimators

z Leads us to confidence intervals