ESTIMATING
with confidence
Confidence INterval
A confidence interval gives an
estimated range of values which is
likely to include an unknown
population parameter, the estimated
range being calculated from a given
set of sample data.
CI: the Basic
The admission directors at the Big City University has
a novel idea. He proposes using the IQ scores of
current students as a marketing tool. So the director
gives the IQ test to an SRS of 50 of the university's
5000 freshmen. The mean IQ score of the sample is
112. What can the director say about the mean score
of the population of all 5000 freshmen?
Supposed we know the
standard deviation σ=15
=112
n=50
N=5 000
σx-bar = 15 / √50 =2.1
=112 σx-bar =2.1
From and SRS of 50
Margin of error: m=2σ=4.2
Conclusion: IQ @ BCU
_
Our sample of 50
freshmen
gave
x
=
112.
The resulting interval is 112 ± 4.2, (107.8,
116.2). We say that we are 95% confident
that the unknown mean IQ score for all BCU
freshmen is between 107.8 and 116.2
1. The interval between 107.8 and 116.2 contains the
true µ.
2. Our SRS was one of the few samples for which x is
not within 4.2 points of the true µ. Only 5% of all
samples give such inaccurate result
Let’s collect several sets of SRS
of 50 students
EXAMPLE
Bianca wanted to know the average
weight of all the female students in
UCLA. She went to UCLA and surveyed
50 female students in random and found
out that their mean weight is 135lbs. She
estimated that the population standard
deviation is about 3.5lbs. With all the
information that she gathered, How can
she estimate the true average weight of
all the female students within 2standard
Confidence INterval and Confidence
Level
A level C confidence interval for a parameter
has 2 points:
•A confidence interval calculated from the
data, usually form
Estimate ± margin of error
•A confidence interval C which gives the
probability that the interval will capture the
true parameter value in repeated samples.
That is, the confidence level is the success
rate for the method
Conditions for Constructing the
Confidence Interval for ℳ
1. The data come from SRS from the
population of interest.
2. The sampling distribution
_
approximately Normal.
of
x
is
3. Individual observations are independent;
when sampling without replacement, the
population size N is at least 10 times the
sample size of n.
AP Test TIP
Common error for students
taking AP Test: Failure to
identify the conditions by
which they are justified in
constructing a confidence
interval.
Most common Confidence
Level
Confidence
Tail Area
level
Z*
90%
0.05
1.645
95%
0.025
1.960
99%
0.005
2.567
Confidence INterval for a populations
σ
MEAN ( is known)
_
X ± z*
σ
√n
Inference toolbox
Step 1: Identify the population of interest and the
parameter you want to draw conclusion about.
Step 2: Choose the appropriate inference
procedure. Verify the conditions for using the
selected procedure.
CI: estimate ± margin of error
Step 4: Interpret your result in the context of the
problem
IQ TEST SCORES
Here are the IQ test scores of 31 7th grade girls in a
midwestern school district:
114 100 104 89 102 91 114 114 103 105 108
130 120 132 111 128 118 119 86 72 111 103
74 112 107 103 98 96 112 112 93
-
Supposed that the standard deviation of IQ
scores in this population is know to be
σ=15.
Construct and interpret a 99% confidence
interval for the mean IQ score in the
population.
SRS: The data came from 7th grade girls in a midwestern school district
Normality: Since the sample came from 31 7th grade
students, we can say that from the Central Limit Theorem
that the sampling distribution of x-bar will be approximately
Normal.
Independence: from the 1st rule of thumb in sample
proportion, where N≧10, a sample size of 310 is more
likely less than the population of all 7th grade female
midwestern students.
The 99% CI for ℳ is:
105.84 ± 2.576 (15/√31) =105.84 ±6.94 =(98.90, 112.78)
With 99% confidence, we estimate the mean IQ score for
all 7th grade girls in the school district to be between 98.90
and 112.78 IQ points
Interpreting Confidence Interval
Interpreting Confidence Interval
When collecting ______
samples repeatedly, we are
________ confident that the
true µ will be contained in
that interval.