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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
TI 84:
ENTER invt DISTR VARS2nd
df) left, toarea( invt
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
TI 83/84:
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