The Practice of StatisticsThird Edition

Chapter 10:Estimating with Confidence

Copyright © 2008 by W. H. Freeman & Company

Calculator Skills• Go to page 630

• Enter the data from the screen tension example into L1

• Press STAT|TESTS|7:ZInterval

– Input is Data

– σ = 43 (given in problem)

– List:L1

– Freq: 1

– C-Level: .9 (90% Confidence Level)

– Calculate (Press Enter)

• What do you see?

Estimating a Population Mean

• How do we construct confidence interval for an unknown µ when we don’t know σ?

• It is unrealistic to assume that you will know σ.

• We must estimate σ from the data even though we are most interested in µ.

• Changes some computations but not the interpretation.

As before, we need to verify three important conditions before we

estimate a population mean

When we do inference in practice, verifying conditions is often a

bit more complicated.

In this setting, x-bar has the Normal distribution with mean µ

and standard deviation σ/√n.

Because we don’t know σ, we estimate it by the sample standard

deviation s. We then estimate the standard deviation of x-bar by

s/√n.

So we are doing away with σ, because it is unrealistic to assume

that we are going to know it and use something much more useful

that we do know.

t Distributions

• When we don’t know σ, we substitute the

standard error s/√n of x-bar for its standard

deviation σ/√n.

• The distribution of the resulting statistic, t,

is not Normal. It is a t distribution. (Before

we were using z.)

• There is a different t distribution for each

sample size n.

t Distributions

• We specify a particular distribution by giving its Degrees of Freedom (df).

• The appropriate df is df = n – 1.

• Why n -1? We are using sample standard deviation s in our calculation and s has n – 1 degrees of freedom.

– We will write a t distribution with k degrees of freedom as t(k).

• We will also refer to standard Normal distribution as the z distribution.

Density curves of t distributions are similar to the Standard Normal Curve.

Symmetric about zero, single peaked and bell shaped.

Spread of a t distribution is a bit greater than Standard Normal curve.

As degrees of freedom k increase the t(k) density curve approaches the N(0,1) curve

ever more closely.

This happens because s estimates σ more accurately as n increases.

So using s in place of σ causes little extra variation when n is large.

Please Note

• The density curve of the t distributions are similar in shape to standard Normal curve.– Symmetric, single-peaked, bell-shaped

• The spread of t distributions is a bit greater than standard Normal curve.– More area in tails and less in the center.

• As the degrees of freedom k increases, the density curve approaches the standard Normal curve more closely.– s estimates σ more accurately as n increases

This is table C from the back of the book. It gives the critical values t* for t

distributions. Degrees of freedom is the left column. Confidence level C is

at the bottom of the table.

What critical value should you use to construct a 95% CI when n = 12?

This one-sample interval is similar in both reasoning and

computational detail to the z-interval from earlier this chapter.

To construct a confidence interval for μ based on a sample from

a Normal population with unknown σ, replace the standard

deviation (σ/√n) of x-bar by its standard error (s/√n) and use the

critical value t* in place of z*

Constructing a One-sample t

Interval for μ

• Construct and interpret a 95% confidence interval for the mean amount of NOX emitted by light duty truck engines. (p. 646)

• 4 Steps – Just like using z*

– Parameter

– Conditions (SRS, Normality, Independence)

– Calculations

– Interpretation

Parameter

• What is the parameter of interest?

– Population is all light duty truck engines of this type.

– We want to estimate μ, the mean amount of the

pollutant NOX emitted for these engines. This is our

parameter.

Conditions

• SRS

– We are told that is the case.

• Normality

– Is the population distribution Normal? How do

we know?

Notice the roughly symmetric shape and the high outliers.

Proceed with caution and examine the impact of outliers

later.

Here is a stem and leaf and box plot of the data. What do you notice?

It is somewhat linear (which we want), but the one high outlier

is very obvious.

Construct a Normal probability plot in your calculator.

Put the data from table 10.2 on page 647 in L2.

Calculations• x-bar = 1.329 grams per mile

• s = .484

• df = 46 – 1 = 45

• No row for 45 df in Table C, so use df = 40 (round down

always)

• Using 40 gives us a wider CI than we need to justify our

given CI

• t* = 2.021 (This is our critical value)

• x-bar ± t*(s/√n) = 1.329 ± 2.021(0.484/√46)

• 1.329 ± 0.144 = (1.185, 1.473)

Interpretation

• We are 95% confident that the true mean of

level of nitrogen oxides emitted by this type

of light-duty engine is between 1.185 and

1.473 grams

• The one-sample t confidence interval has

the form:

• Estimate ± t*(SEestimate)

• Where SE stands for Standard Error (s/√n)

Calculator Skills

• STAT|TESTS|8:TInterval

– Input: Data

– List: L2

– Freq: 1

– C-Level: .95

– Calculate (press Enter)

Note: these are 1-sample t-intervals that we are

doing now, and we were doing 1-sample z-

intervals before.

Assignment

• Exercises 10.27, 10.28, 10.31

• Read pages 651 – 657

• Watch:https://youtu.be/-7nxSAOgAQ4?list=PLkIselvEzpM7N8zVRRUl7V8aTdoTsJ919

Get comfy doing one sample t-intervals on your

calculator.