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ARE YOU HERE?
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Yes, and I’m
ready to learn
Yes, and I need a
nap
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APPLE PROBLEM

When a truck load of apples arrives at a packing plant,
a random sample of 125 is selected and examined for
bruises, discoloration, and other defects.
The whole truckload will be rejected if more than 5% of
the sample is unsatisfactory.
 Suppose that in fact 9% of the apples on the truck do
not meet the desired standard.
 What is the probability that the shipment will be
accepted anyway.

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STANDARD DEVIATION

Both of the sampling distributions we’ve looked at are
Normal.


For proportions
For means
pq
n
SD  pˆ  
SD  y  

n
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WHAT IS THE PROBABILITY THAT THE
SHIPMENT WILL BE ACCEPTED ANYWAY?
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2.
3.
4.
5.
0.062
1-0.062
0
1
-1.54
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0%
0%
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0%
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5.
STANDARD DEVIATION VS. STANDARD
ERROR


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
We don’t know p, μ, or σ, we’re stuck, right?
Nope. We will use sample statistics to estimate
these population parameters.
Sample statistics are notated as: s,
Whenever we estimate the standard deviation of
a sampling distribution, we call it a standard
error.
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STANDARD ERROR

For a sample proportion, the standard error is
SE  pˆ  

ˆˆ
pq
n
For the sample mean, the standard error is
s
SE  y  
n
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EXCITING STATISTICS ABOUT ISU
STUDENTS -2011 DATA

69.1% of sexually active students use condoms

American College Health Association

n=272
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A CONFIDENCE INTERVAL
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A CONFIDENCE INTERVAL

By the 68-95-99.7% Rule, we know
about 68% of all samples will have p
ˆ ’s within 1 SE of p
 about 95% of all samples will have p
ˆ ’s within 2 SEs of p
 about 99.7% of all samples will have p
ˆ ’s within 3 SEs of p

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CERTAINTY VS. PRECISION
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CERTAINTY VS. PRECISION


The choice of confidence level is somewhat
arbitrary, but keep in mind this tension between
certainty and precision when selecting your
confidence level.
The most commonly chosen confidence levels are
90%, 95%, and 99% (but any percentage can be
used).
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WHAT DOES “95% CONFIDENCE”
REALLY MEAN?


Each confidence interval uses a sample statistic
to estimate a population parameter.
But, since samples vary, the statistics we use,
and thus the confidence intervals we construct,
vary as well.
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WHAT DOES “95% CONFIDENCE”
REALLY MEAN? (CONT.)

The figure to the
right shows that
some of our
confidence intervals
capture the true
proportion (the green
horizontal line),
while others do not:
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SALES PROBLEM


A catalog sales company promises to deliver
orders placed on the Internet within 3 days.
Follow-up calls to a few randomly selected
customers show that a 95% CI for the proportion
of all orders that arrive on time is 81% ± 4%
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WHICH OF THE FOLLOWING STATEMENTS IS
CORRECT?
1.
2.
3.
4.
Between 77% and 85% of all orders arrive on time.
One can be 95% confident that the true population
percentage of orders place on the Internet that arrive
within 3 days is between 77% and 85%
One can be 95% confident that all random samples of
customers will show that 81% of orders arrive on time
95% of all random samples of customers will show
that between 77% and 85% of orders arrive on time.
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ONE-PROPORTION Z-INTERVAL
When the conditions are met, we are ready to find the
confidence interval for the population proportion, p.
 The confidence interval is

pˆ  z  SE  pˆ 

where

ˆˆ
SE( pˆ )  pq
n
The critical value, z*, depends on the particular
confidence level, C, that you specify.
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Z* IS THE CRITICAL VALUE
80%  z*=1.282
 90% z*=1.645
 95% z*=1.96
 98%z*=2.326
 99% z*=2.576

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CRITICAL VALUES (CONT.)

Example: For a 90% confidence interval, the
critical value is 1.645:
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BIASED SURVEY PROBLEM

Often, on surveys there are two ways of asking
the same question.
1) Do you believe the death penalty is fair or
unfairly applied?
 2) Do you believe the death penalty is unfair or
fairly applied?

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BIASED SURVEY PROBLEM

Survey
1) n=597
 2) n=597


For the second phrasing, 45% said the death
penalty is fairly applied.
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SUPPOSE 54% OF THE RESPONDENTS IN
SURVEY #1 SAID THE DEATH PENALTY WAS
FAIRLY APPLIED. DOES THIS FALL WITHIN A
95% CONFIDENCE INTERVAL FOR SURVEY #2?
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2.
Yes, it falls within my CI
No, it does not fall within my CI
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MARGIN OF ERROR: CERTAINTY VS. PRECISION

The more confident we want to be, the larger our
z* has to be

But to be more precise (i.e. have a smaller ME
and interval), we need a larger sample size, n.

We can claim, with 95% confidence, that the
interval p
ˆ  2SE( pˆ ) contains the true population
proportion.


The extent of the interval on either side of
the margin of error (ME).
pˆ is called
In general, confidence intervals have the form
estimate ± ME.
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MARGIN OF ERROR - PROBLEM



Suppose the truth is that 56% of ISU student
drink every weekend.
We want to create a 95% confidence interval, but
we also want to be as precise as possible.
How many people should we sample to get a ME
of 1%?
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HOW MANY PEOPLE SHOULD WE SAMPLE
TO GET A ME OF 1%?
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2.
3.
4.
1,000
Between 1,000 and 4,000
Between 4,000 and 8,000
Between 8,000 and 16,000
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UPCOMING WORK

Quiz #4 in class today

HW #8 due Sunday

Part 3 of Data Project due Oct. 28th