AP Statistics Section 10.2 A CI for Population Mean When is Unknown.

AP Statistics Section 10.2 A

CI for Population Mean When is Unknown

In Section 10.1B, we constructed a confidence interval for the

population mean when we knew the population standard deviation . It is extremely

unlikely that we would actually know the population standard

deviation, however.

In this section, we will discover how to construct a confidence

interval for an unknown population mean when we don’t know the standard deviation . We will do

this by estimating from the data.

This need to estimate with s introduces additional error into our

calculations. To account for this, we will use a critical value of t*

instead of z* when computing our confidence interval.

Note the following properties of a t distribution:

The density curves of the t distributions are similar in shape to

the standard Normal, or z, distribution (i.e.

zeroat centered and shaped-bell

Unlike the standard Normal distribution, there is a different t distribution for each sample size n. We specify a particular t

distribution by giving its __________________ ( _____ ).

When we perform inference about using a t distribution, the appropriate degrees of

freedom is equal to ______. We will write the t distribution with k degrees of

freedom as _____.

df freedom of degrees

1-n

t(k)

The spread of the t distributions is slightly greater than that of the z distribution. The t

distributions are less concentrated around the mean and have more probability in the tails. This is what accounts for the increased error in using

s instead of .

As the degrees of freedom increase, the t curve approaches the standard Normal curve ever more closely. This happens

because s approximates more accurately as the sample size increases.

Table B, gives the values of t* for various degrees of freedom and various upper-tail probabilities. When the actual degrees of freedom does not appear in Table B, use the largest degrees of freedom that is less

than your desired degrees of freedom.

Example: Determine the appropriate value of t* for a confidence interval for with the given

confidence level and sample size. a) 98% with n = 22

21122 df

518.2t

.01 .98 01.

Example: Determine the appropriate value of t* for a confidence interval for with the given

confidence level and sample size. b) 90% with n = 38

37138 df

697.1t

.05 .90 05.

tableon the 30df usemust

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

estimate a population mean.

SRS: Our data are a SRS of size n from the population of interest or

come from a randomized experiment. This condition is very

important.

Normality of : The population has a Normal distribution or: Use t procedures if sample data appears roughly Normal.: The t procedures can be used except in the presence of outliers or strong skewness in the sample data. The t procedures are robust.: The t procedures can be used even for clearly skewed distributions. However, outliers are still a concern. You may still refer to the Central Limit Theorem in this situation.

x

15n

15n

30n

Independence: The method for calculating a confidence interval assumes that individual observations are independent. To keep the calculations reasonably accurate when we sample without replacement from a finite

population, we should verify that the population size is at least

_______________________(________).size sample the times10 nN 10

n

stx

Example: A number of groups are interested in studying the auto exhaust emissions produced by motor

vehicles. Here is the amount of nitrogen oxides (NOX) emitted by light-duty engines (grams/mile) from a

random sample of size n = 46. Construct and interpret a 95% confidence interval for the mean amount of NOX

emitted by light-duty engines of this type.

Parameter: The population of interest is

____________________.

We want to estimate , the ____________________________.

enginesduty -light

emitted NOX ofamount mean

Conditions: Since we do not know , use ______________________ SRS:

Normality of :

Independence:

interval t sample-one a

46.n size of sample random a from comes Data

Normal.ely approximat is x ofon distributi

theand applies CLT the46,nWith

10n.N that so 460least at is population the

assumemust wet,replacemen w/osampling Since

x

Calculation:

46

484.

329.1

n

s

x

021.2

45146

t

df

)473.1,185.1(

144.329.146

484.021.2329.1

Interpretation:

.grams/mile 1.473 and 1.185between is enginesduty -lightfor

emissions NOX ofmean that theconfident 95% are We

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Interval-T Tests STATS

Standard ErrorWhen the standard deviation of a statistic, i.e. , is estimated from the data, the result is called

the standard error of the statistic.

p̂or x

Some textbooks simply refer to standard error as the standard

deviation of the sampling distribution, , whether it is

estimated from the data or not.p̂or x