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

AP Statistics Section 8.3C
CI for Population Mean When  is
Unknown
In Section 8.3A, 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.
bell - shaped and centered at zero
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
__________________
degrees of freedom ( _____
df ).
When we perform inference about  using
a t distribution, the appropriate degrees of
freedom is equal to ______.
n - 1 We will write
the t distribution with k degrees of
freedom as _____.
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
.01
.98
.01
df  22  1  21

t  2.518
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
.05
.90
.05
df  38  1  37
must use df  30 on the table

t  1.697
TI 84:
2 nd VARS DISTR invt ENTER
invt (area to left, df)
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 x : The population has a Normal
distribution or
: n  15 Use t procedures if sample data appears
roughly Normal.
: n  15 The t procedures can be used except in the
presence of outliers or strong skewness in the sample
data. The t procedures are robust.
: n  30 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.
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
_______________________(________).
10 times the sample size N  10n
x t

s
n
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 ____________________.
light - duty engines
We want to estimate , the
mean amount of NO x emitted
____________________________.
Conditions: Since we do not know  , use
______________________
a one - sample t interval
SRS: Data comes from a random sample of size n  46.
Normality of x : With n  46, the CLT applies and the
distribution of x is approximately Normal.
Independence:
The population of light duty engines is at least
10  46 or 460.
Calculation:
x  1.329
s  .484
n  46
x t
*
s
n
df  46  1  45

t  2.021
.484
1.329  2.021
46
1.329  .144
(1.185,1.473)
Interpretation:
We are 95% confident that the mean of NO x emissions
for light - duty engines is between 1.185 and 1.473 grams/mile.
TI 83/84:
STATS Tests T - Interval
Standard Error
When the standard deviation of a
statistic, i.e.  x or  p̂ , is estimated
from the data, the result is called
the standard error of the statistic.
Some textbooks simply refer to
standard error as the standard
deviation of the sampling
distribution, x or  p̂ , whether it is
estimated from the data or not.