1 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison...

1Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICS

Section 6-3 Estimating a Population Mean: Small Samples


If 1) n 302) The sample is a simple random sample.

3) The sample is from a normally distributed population.

Case 1 ( is known): Largely unrealistic; Use methods from 6-2

Case 2 (is unknown): Use Student t distribution

Small SamplesAssumptions


Student t Distribution

If the distribution of a population is essentially normal, then the distribution of

t =x - µ

sn


Student t Distribution

If the distribution of a population is essentially normal, then the distribution of

is essentially a Student t Distribution for all samples of size n.

is used to find critical values denoted by

t =x - µ

sn

t/ 2


Using the TI Calculator

Find the “t” score using the

InvT command as follows:

2nd - Vars – then

invT( percent, degree of

freedom )


DefinitionDegrees of Freedom (df )

corresponds to the number of sample values that can vary after certain restrictions have imposed

on all data values


DefinitionDegrees of Freedom (df )

corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values

df = n - 1in this section


DefinitionDegrees of Freedom (df ) = n - 1

corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values

Any

#Specific

#

so that x = a specific number

Any

#

n = 10 df = 10 - 1 = 9

Any

#Any

#Any

#Any

#Any

#Any

#Any

#


Margin of Error E for Estimate of Based on an Unknown and a Small Simple Random

Sample from a Normally Distributed Population


Margin of Error E for Estimate of Based on an Unknown and a Small Simple Random

Sample from a Normally Distributed Population

Formula 6-2

where t/ 2 has n - 1 degrees of freedom

nE = ts

2


Confidence Interval for the Estimate of E

Based on an Unknown and a Small Simple Random Sample from a Normally Distributed Population




x - E < µ < x + E




x - E < µ < x + E

where E = t/2 ns




x - E < µ < x + E

where E = t/2 ns

t/2 found using invT


Properties of the Student t Distribution

There is a different distribution for different sample sizes

It has the same general bell shape as the normal curve but greater variability because of small samples.

The t distribution has a mean of t = 0

The standard deviation of the t distribution varies with the sample size and is greater than 1

As the sample size (n) gets larger, the t distribution gets closer to the normal distribution.


Student tdistributionwith n = 3

Student t Distributions for n = 3 and n = 12

0

Student tdistributionwith n = 12

Standardnormaldistribution

Figure 6-5


Figure 6-6

Using the Normal and t Distribution


Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.)



x = 26,227

s = 15,873

= 0.05/2 = 0.025



x = 26,227

s = 15,873

= 0.05/2 = 0.025t/2 = 2.201

E = t2 s = (2.201)(15,873) = 10,085.29

n 12



x = 26,227

s = 15,873

= 0.05/2 = 0.025t/2 = 2.201

E = t2 s = (2.201)(15,873) = 10,085.3

n 12

x - E < µ < x + E



x = 26,227

s = 15,873

= 0.05/2 = 0.025t/2 = 2.201

E = t2 s = (2.201)(15,873) = 10,085.3

n 12

26,227 - 10,085.3 < µ < 26,227 + 10,085.3

x - E < µ < x + E



x = 26,227

s = 15,873

= 0.05/2 = 0.025t/2 = 2.201

E = t2 s = (2.201)(15,873) = 10,085.3

n 12

26,227 - 10,085.3 < µ < 26,227 + 10,085.3

$16,141.7 < µ < $36,312.3

x - E < µ < x + E



x = 26,227

s = 15,873

= 0.05/2 = 0.025t/2 = 2.201

E = t2 s = (2.201)(15,873) = 10,085.3

n 12

26,227 - 10,085.3 < µ < 26,227 + 10,085.3

x - E < µ < x + E

We are 95% confident that this interval contains the average cost of repairing a Dodge Viper.

$16,141.7 < µ < $36,312.3

1 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison...

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