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More about the Sampling More about the Sampling Distribution of the Sample Distribution of the Sample

Mean Mean

and and

introduction to the introduction to the

t-distributiont-distribution

Presentation 3

22

Brief Review of Sampling Brief Review of Sampling DistributionsDistributions

If the sample is large enough, If the sample is large enough, npnp and and n(1-p)n(1-p) >5, >5, thenthen

size sample totaltrait with sample the inelement ofnumber

ˆ p

Sampling Distribution of Sample ProportionSampling Distribution of Sample Proportion:: Each member of the population has a trait of interest Each member of the population has a trait of interest

with probability with probability p p (population proportion). (population proportion). Suppose a random sample of size Suppose a random sample of size nn is obtained form is obtained form

the population.the population. The sample proportion The sample proportion p-p-hat is a logical estimator of hat is a logical estimator of

p,p,

),(~)(

ˆ)(,~ˆ

..

101

1N

npp

ppZ

npp

pNpapproxapprox

33

Sampling Distribution of Sample Sampling Distribution of Sample MeanMean

Let Let XX be a random variable and the statistic be be a random variable and the statistic be the sample mean of the sample mean of XX in a random sample of size in a random sample of size nn. .

We examine the sampling distribution of in the We examine the sampling distribution of in the following three scenarios:following three scenarios:

1.1. When When XX is a normal random variable, is a normal random variable, E(X)=E(X)=µ µ and s.d.(X)= and s.d.(X)= σσ are both known. are both known.

2.2. When When XX is not a normal random variable, is not a normal random variable, E(X)=E(X)=µ and s.d.(X)= µ and s.d.(X)= σσ are both known are both known and the and the sample size is large, sample size is large, nn ≥ 30.≥ 30.

3.3. When When XX is a normal random variable, is a normal random variable, E(X)=E(X)=µ is µ is known, s.d.(X)=known, s.d.(X)= σσ is unknown and is unknown and the sample the sample size is large, size is large, nn ≥ 30.≥ 30.

So far we have seen the first two casesSo far we have seen the first two cases. .

44

Case 1: Case 1:

),(~,~)(

)(

10

30

22 N

n

Xn

NX

n

XVar

XEapproxapprox

Case 2: Case 2:

Case 3:Case 3:

1

ntns

X

X

sX

XE

~

r.v. normal

)(s.d. sample

)(

),(~,~

r.v. normal

)(

)(

102

2 Nn

Xn

NX

X

XVar

XE

, where t, where tn-1n-1 denotes the denotes the t-distribution with t-distribution with n n -1 -1 degrees of freedom. degrees of freedom.

55

Properties of the t-distributionProperties of the t-distribution There are infinitely many There are infinitely many t-t-distributions, each distributions, each

characterized by one parameter, the degrees of characterized by one parameter, the degrees of freedom. freedom.

The degrees of freedom are positive integers, e.g. The degrees of freedom are positive integers, e.g. tt11, t, t2 2 , t, t33,…, t,…, t1010,…,…

Random variables with t-distribution are Random variables with t-distribution are continuous.continuous.

The density curve of aThe density curve of a t t - distribution is - distribution is symmetric, bell-shaped and centered at zero symmetric, bell-shaped and centered at zero (similar to the standard normal curve).(similar to the standard normal curve).

There are tables for the probabilities related with There are tables for the probabilities related with a a t – t – random variable. In will see how to use them random variable. In will see how to use them later in the course. later in the course.

As the degrees of freedom increase, the variance As the degrees of freedom increase, the variance of the of the t t -random variable decreases, i.e. the -random variable decreases, i.e. the density curve is less spread, and actually it density curve is less spread, and actually it approaches the standard normal density.approaches the standard normal density.

66

Properties of the t-distributionProperties of the t-distribution

Z~N(0,1)

t10

t3

t1