Download AP Statistics Section 10.1 B CI for Population Mean When is Known

AP Statistics Section 8.3A
CI for Population Mean When  is
Known
Before calculating a confidence
interval for  or p, there are three
important conditions that you
must check:
1. The data should come from an ______
SRS of the population.
a) Often times, the problem will simply state the sample is
random but not specifically say it is an SRS. While the
calculations we use technically require an SRS, the AP
exam allows to simply state that you have a random
sample.
b) If the data does not come from a random sample of the
population,
the results may not generalize to the population.
c) The margin of error in a confidence interval covers only
chance variation due to random sampling. It does not
account for mistakes in our sampling method such as
undercoverage.
2. The sampling distribution of x or p̂ must be at least approximately Normal.
For means:
a) If the population distribution is Normal, then the distribution of x is
Normal.
b) If the population distribution is not Normal, then
as long as the sample size is large enough (n  30) the
distribution of x will be approximately Normal by CLT
c) If neither a) nor b) is appropriate, look at the sample data. If the sample
data does not show any striking deviations from Normality (outliers or strong
skewness), we will assume that the population distribution is at least
approximately Normal and therefore the distribution of x is approximately
Normal.
d) If the sampling distribution is not at least approximately Normal, then
the resulting confidence interval may not be accurate
3. The individual observations in the random
sample must be independent.
a) Since we almost always sample without
replacement, we need to verify that
the population is at least 10 times as large as the sample
N  10n
(_________)
b) If the individual observations are not
independent, our calculations may not be
accurate.
xz


n
Example: Find the value of z* for the following
confidence levels.
88%
b) 95%
.06
z
.025
.88
*
 0
 1
z
z  1.555
z

invNorm(.94,0,1)

.95
invNorm(.975,0,1)
z   1.960
These z-scores that “mark off” a
specified area under the Standard
Normal curve are often called
critical values.
The confidence levels at the right and
their corresponding upper critical
value are so common that they are
worth memorizing so that you will not
have to take the time to find them
each time.
Confidence Level
90%
95%
99%
Tail Area
0.05
0.025
0.005
z*
1.645
1.960
2.576
When constructing a confidence interval you should use the
tools from the Inference Toolbox.
Parameter
Step 1: _______________ Identify the population of interest
and the parameter you want to draw conclusions about.
Conditions
Step 2: _______________ Identify the appropriate inference
procedure and verify the conditions for using it.
ns Carry out the inference procedure:
Step 3: Calculatio
_______________
Step 4: ___________________
Interpretations State your conclusions in the
context of the problem.
Example: A manufacturer of high-resolution video
terminals must control the tension on the mesh of fine
wires that lies behind the surface of the viewing screen.
Too much tension will tear the mesh and too little will
allow wrinkles. The tension is measured by an electrical
device with output readings in millivolts (mV). Some
variation is inherent in the production process. Careful
study has shown that when the process is operating
properly, the standard deviation of the tension readings
is   43 mV.
Construct and interpret a 90%
confidence interval for the mean
tension of all such screens.
Parameter: The population of interest is
high resolution video terminals.
We want to estimate , the
mean tension for the wire mesh in
these screens.
Conditions: Since we know ____
 use the CI for a
population mean where  is known.
SRS:
Data comes from a random sample of 20 screens
Normality of x : Since we do not know if the
population distribution is Normal….
CLT?
NO, with n  20 the sample size is too small
Boxplot
Normal probability plot?
Sample appears approx. Normal so we will assume
the population is approx. Normally distributed.
Independence:
Since we sampled without replacement, we must
assume the population of such terminals is at least
10(20)  200
Calculations:

For C  90%, z  1.645
x  306.32
xz


43
 306.32  1.645
 306.32  15.82
n
20
(290.5,322.14)
Interpretation:
We are 90% confident that the mean tension
of the mesh of fine wires in the video terminals
is between 290.5 and 322.14 millivolts.
TI-83/84: STAT Tests
7 : z - interval