The t Distributions Sections 20.1, 20people.hsc.edu/faculty-staff/robbk/math121/lectures... · Robb...

The t DistributionsSections 20.1, 20.2

Lecture 35

Robb T. Koether

Hampden-Sydney College

Thu, Mar 24, 2016

Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 1 / 18

Outline

1 The Central Limit Theorem

2 Substituting s for σ

3 The t Distributions

4 Comparison of t to z

5 Assignment


Outline





5 Assignment


The Central Limit Theorem

Theorem (The Central Limit Theorem)For any population, let its mean and standard deviation be µ and σ,respectively. let x be the sample mean of samples of size n. Then xhas an approximately normal distribution with mean µ and standarddeviation σ/

√n.

If the sample size is large enough (n ≥ 30), then theapproximation is good enough for applications.


The Central Limit Theorem

It follows from this theorem that the statistic z, defined as

z =x − µσ/√

n

has an approximately normal distribution.


Special Case

Theorem (Special Case of the CLT)If the population is normal, then for all sample sizes, no matter howsmall, x has an exactly normal distribution with mean µ and standarddeviation σ/

√n.


Special Case

It follows from this theorem that if the population is normal, thenthe statistic

z =x − µσ/√

n

has an exactly normal distribution for any sample size n.


Outline





5 Assignment


Substituting s for σ

In practice we rarely know the value of σ.However, we do know the value of s and s is an estimator of σ.What is the effect of replacing σ with s in the formula

x − µσ/√

n?

s is a variable and σ is a constant.The effect is to increase the variability, and therefore theuncertainty, of the value of the statistic.


Outline





5 Assignment


The t Statistic

It turns out that the statistic

x − µs/√

n

does not have a normal distribution, even when sampling from anormal population, unless n is fairly large (n > 100).So we name the statistic t instead of z:

t =x − µs/√

n.

Furthermore, the exactly distribution of t is different for differentsample sizes.(As n increases, t tends towards z.)


The t Distributions

Definition (The t Distributions)If we draw a simple random sample of size n from a normal populationwith mean µ and standard deviation σ, then the t statistic with n − 1degrees of freedom is

t =x − µs/√

n


Outline





5 Assignment


Comparison of t to z

-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4

2 degrees of freedom



-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4



At n = 10, the difference between t and z does not appear to begreat.However, p-values usually involve the tails.How do the upper tails of t and z compare?


Comparison of the Upper Tails

2.5 3.0 3.5 4.0 4.5

0.01

0.02

0.03

0.04




2.5 3.0 3.5 4.0 4.5

0.01

0.02

0.03

0.04



Outline





5 Assignment


Assignment

AssignmentRead Sections 20.1, 20.2.Apply Your Knowledge: 1, 3, 4.Check Your Skills: 17, 18, 19.


The t Distributions Sections 20.1, 20people.hsc.edu/faculty-staff/robbk/math121/lectures... · Robb...

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