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Section 10.1
Estimating with
Confidence
AP Statistics
January 2013
1
np  10 or nq  10?
Use Binomial distribution tools.
Sample Proportions?
Make sure the population size  10n
pq
n
so you may use  pˆ 
np  10 and nq  10?
Use Normal distribution tools.
Is the population distribution normal?
Use Normal distribution tools.
Sample Means?
Make sure the population size  10n
so you may use  x 

n
Is the shape of population distribution
unknown or distinctly nonnormal?
If n  30, the Central Limit Theorem
applies so you may use Normal distribution tools.
2
Otherwise, you need other tools.
An introduction to statistical
inference



Statistical Inference provides methods for
drawing conclusions about a population from
sample data.
In other words, from looking a sample, how
much can we “infer” about the population.
We may only make inferences about the
population if our samples unbiased. This
happens when we get our data from SRS or
well-designed experiments.
3
Example
A SRS of 500 California high school
seniors finds their mean on the SAT Math
is 461. The standard deviation of all
California high school seniors on this test
111.
 What can you say about the mean of all
California high school seniors on this
exam?

4
Example (What we know)
Data comes from SRS, therefore is
unbiased.
 There are approximately 350,000
California high school seniors.
350,000>10*500. We can estimate

111



 4.5
sigma-x-bar as
n
500
 The sample mean 461 is one value in the
distribution of sample means.

x
5
Example (What we know)
The mean of the distribution of sample
means is the same as the population
mean.
 Because the n>30, the distribution of
sample means is approximately normal.
(Central Limit Theorem)

6
Our sample is just one value in a
distribution with unknown mean…
7
Confidence Interval

A level C confidence interval for a
parameter has two parts.
 An
interval calculated from the data, usually in
the form (estimate plus or minus margin of
error)
 A confidence level C, which gives the long
term proportion that the interval will capture
the true parameter value in repeated samples.
8
9
Conditions for Confidence Intervals
the data come from an SRS or well
designed experiment from the population
of interest
 the sample distribution is approximately
normal

10
11
Confidence Interval Formulas
CI  x  z
CI  x  z
*

*

n
,x  z
*

n
n
*
where z is the upper p critical value
12
Using the z table…
Confidence
level
Tail Area
z*
90%
.05
1.645
95%
.025
1.960
99%
.005
2.576
13
Four Step Process
(Inference Toolbox)

Step 1 (Pop and para)


Define the population and parameter you are
investigating
Step 2 (Conditions)

Do we have biased data?


Do we have independent sampling?


If SRS, we’re good. Otherwise PWC (proceed with caution)
If pop>10n, we’re good. Otherwise PWC.
Do we have a normal distribution?

If pop is normal or n>30, we’re good. Otherwise, PWC.
14
Four Step Process
(Inference Toolbox)

Step 3 (Calculations)

Find z* based on your confidence level. If
you are not given a confidence level, use
95%
 Calculate CI.

Step 4 (Interpretation)

“With ___% confidence, we believe that the
true mean is between (lower, upper)”
15
Confidence interval behavior

To make the margin
of error smaller…
 make
z* smaller,
which means you have
lower confidence
 make n bigger, which
will cost more
margin of error  z
*

n
16
Confidence interval behavior

If you know a
particular confidence
level and ME, you can
solve for your sample
size.
margin of error  z
*

n
17
Example


Company management
wants a report screen
tensions which have
standard deviation of 43
mV. They would like to
know how big the sample
has to be to be within 5
mV with 95% confidence?
You need a sample size
of at least 285.
ME  z
*

n
43
5  1.96
n
43
n  1.96
5
2
43 

n  1.96   284.12
5 

18
Mantras

“Interpret 80% confidence interval of (454,467)”
 With
80% confidence we believe that the true mean of
California senior SAT-M scores is between 454 and
467.

“Interpret 80% confidence”
 If
we use these methods repeatly, 80% of the time our
confidence interval captures the true mean.
 Probability
19
Assignment

Exercises 10.1 to 10.8
20