AP Statistics
Hamilton/Mann
Introduction
How long can you expect a AA battery to last?
 What proportion of college undergraduates have
engaged in binge drinking?
 How long can you expect to spend working on
your AP Statistics homework?
 Is caffeine dependence real?
 It is not practical to determine the life of all AA
batteries, to ask all college undergraduates about
their drinking habits, to ask every AP Statistics
students how long they spend on homework or to
talk to every potential caffeine dependent
individual in the population.

So What Do We Do
Instead, we select a sample of individuals to
represent the population, and we collect data from
those individuals.
 We want to infer from the sample data some
conclusion about the population.
 That is the goal of statistical inference.


We cannot be certain that our results are correct
because a different sample might lead to different
conclusions.
So What Do We Do (cont)
Statistical inference uses the language of
probability to express the strength of our
conclusions.
 Probability allows us to take chance variation into
account and correct our judgment by calculation.
 In this chapter, we will learn about confidence
intervals for estimating the value of a population
parameter.
 This technique uses the ideas of sampling
distributions to report what would happen if we
repeated the inference method many times.

Inference
The methods of formal inference require the longrun regular behavior that probability describes.
 Inference is most reliable when the data are
produced by a properly randomized design.
 When you use statistical inference, you are acting
as if the data are a random sample or come from
a randomized experiment. If this is not true, your
conclusions may be open to challenge.
 Formal inference cannot remedy problems in data
collection (such as voluntary response samples
and uncontrolled experiments).
 Proceed to formal inference only when you are
satisfied that the data deserve such analysis.

Inference

We will make the problems simpler to begin with.
We will pretend like we know the standard
deviation for the population that we are interested
in, even though we do not know its mean. We will
later remove this unrealistic requirement and learn
how to handle situations in which there is no way
we could know the standard deviation of the
population.
Confidence Intervals: The Basics
Section 1 HW: 10.1, 10.2, 10.3, 10.4,
10.7, 10.8, 10.10, 10.11, 10.14, 10.15,
10.16, 10.19
An Example to Begin

Colleges and universities attempt to attract
qualified students by putting out information such
as average GPA, average SAT scores, class
ranks, etc. At Big City University, the admissions
director had a novel idea. He proposed that the
school use the IQ scores of current students as a
marketing tool. The school agrees to give him
enough money to administer an IQ test to 50
students. He selects an SRS of 50 of the 5000
freshmen. The mean IQ score from the sample is
So what can the director say about the
mean score of the population of all 5000
freshmen?
What About the Example
Is the mean IQ score of all Big City University
freshmen exactly 112? Probably not!
 The law of large numbers tells us though that the
sample mean from a large SRS will be close to
the unknown population mean
 Because
we guess that is “somewhere
around 112.” The question though is “How close
to 112 is likely to be?”
 To answer this question, we ask another: How
would the sample mean vary if we took many
samples of 50 freshmen from this same
population?

Reminder

The sampling distribution of describes how the
values of vary in repeated samples. Let’s recall
some facts about the sampling distribution.
 The mean of the sampling distribution of
is the same
as the unknown mean of the entire population.
 The standard deviation of
is given by
where
is
the standard deviation of the population of interest.
 The Central Limit Theorem tells us that if our sample
size is large enough (30 or more), then the sampling
distribution will be close to Normal.
Reasoning of Statistical Estimation
To estimate the unknown population mean
use
the mean of our random sample.
2. Although
is an unbiased estimate of
it will
rarely be exactly equal to
so our estimate has
some error.
3. In repeated samples, the values of follow
(approximately) a Normal distribution with mean
and standard deviation
4. The 95 part of the 68-95-99.7 rule for Normal
distributions says that in about 95% of all
samples, the mean will be within two standard
deviations of the population mean
1.
Reasoning of Statistical Estimation (cont.)
5.
6.
Whenever is within two standard deviations of
is within two standard deviations of
This
happens in about 95% of all possible samples.
So the unknown lies between
and
in about 95% of all samples.
If we estimate that
lies somewhere in the
interval from
to
we would be
calculating this interval using a method that
“captures” the true
in about 95% of all
samples.
The Big Idea
The big idea is that the sampling distribution of
tells us how big the error is likely to be when we
use to estimate
 So let’s return to our example of Big City
University.

Big City University
Suppose that the standard deviation of IQ scores
for all freshmen at Big City University is
 Since our sample size was 50 and population is
more than 10 times our sample size,

Also, since our sample is large, the CLT tells us
that our sampling distribution is close to Normal
and can be approximated by
 So we can conclude that the would be within two
standard deviations for 95% of all samples.
Therefore, in 95% of samples would be within
the interval

Big City University (cont.)

The interval
can be written as

There are only two possibilities:
1. The interval between 107.76 and 116.24 contains the
true
2. Our SRS was one of the few samples for which is not
within 4.24 points of the true
Only 5% of all samples
give such inaccurate results.

We cannot know if our sample is one of the 95%
for which the interval
catches or
whether it is one of the unlucky 5%.
Big City University (cont.)
The statement that we are “95% confident” that the
unknown lies between 107.76 and 116.24 is
shorthand for saying, “We got these numbers by a
method that gives correct results 95% of the time.”
 The interval of numbers
is called a 95%
confidence interval for The confidence level is 95%.
Like most confidence intervals we will meet, this one
has the form


The estimate is our best guess for the value of the
unknown parameter. The margin of error
shows
how accurate we believe our guess is, based on the
variability of the estimate. This is a 95% confidence
interval because it catches the unknown in 95% of
all possible samples.

This is the basic idea behind confidence intervals.
95% of the samples we select should allow us to
create a confidence interval in which the true
value of the parameter will appear!
We get to choose (or we are told) the confidence
level. This level is usually 90% or higher because
we want to be quite sure of our conclusions.
 We will use C to stand for our confidence level in
decimal form. For example, a 95% confidence
level corresponds to
 Let’s see another explanation of Confidence
Intervals!


This is another way of thinking of confidence
intervals. Some intervals will not contain the true
population parameter, but most will!
Let’s Look at Some Together

10.5 and 10.6
Confidence Interval for a Population
Mean (When is Known)

1.
Now we are going to learn to calculate a level C
confidence interval for the unknown mean of a
population. The calculation of the interval
depends on three important conditions.
SRS – Our method of calculation assumes that
the data comes from an SRS of size n from the
population of interest. Other types of random
samples (stratified or cluster) may give a better
representation of the population than an SRS, but
they would require more complex calculations
than the ones we’ll use.
Confidence Interval for a Population
Mean (When is Known)
2.
Normality – The construction of the interval
depends on the fact that the sampling distribution
of the sample mean is at least approximately
Normal. This distribution is exactly Normal if the
population distribution is Normal. When the
population distribution is not Normal, the Central
Limit Theorem tells us that the sampling
distribution of will be approximately Normal if n
is sufficiently large (say, at least 30).
Confidence Interval for a Population
Mean (When is Known)
3.
Independence – The procedure for calculating a
confidence interval assumes that individual
observations are independent. Independent
observations are required for us to use the
formula
We are safe if we sample with
replacement from the population. We rarely do
this though. To keep our calculations reasonably
accurate when we sample without replacement
from a population, we should sample no more
than 10% of the population. In other words, our
population must be at least 10 times larger than
our sample.
Conditions Summarized

Be sure to check that these conditions for
constructing a confidence interval for are
satisfied before you perform any calculations.
Let’s Find the Critical Value for 80%

We need to find out how many standard deviations
away from the mean to go in order to catch 80% of
the information. In other words, we are leaving
out 20%. Since the Normal curve is symmetric,
we are leaving out the top 10% and the bottom
10%. So our critical value z* would be the z-value
where 90% of the data is less than it. Looking in
the Normal table, we find that z* is 1.28. There is
an area of 0.8 under the standard Normal curve
between -1.28 and 1.28.
This can be seen in the
graph on the next slide.

The figure above shows the general situation for
any confidence level C. If we catch the central
area C, the leftover tail area is 1-C, or (1-C)/2 for
each of the upper and lower tails. You can find z*
for any C by searching Table A.
Common Confidence Levels

Confidence
Level
Tail Area
z*
90%
0.05
1.645
95%
0.025
1.960
99%
0.005
2.576
Notice that for 95% confidence we use z* = 1.960.
This is more exact than the approximate value of
z* = 2 given by the 68-95-99.7 rule. Values of z*
that mark off a specified area under the standard
Normal curve are often called critical values of
the distribution.
Thinking for a Level C Confidence Interval
Under any normal curve, the area between the
point z* standard deviations below its mean and
the point z* standard deviations above its mean is
C.
 The standard deviation of the sampling distribution
of is
and its mean is the population mean
So there is probability C that the observed sample
mean takes a value between
 Whenever this happens, the population mean
is
contained between
 That is our confidence interval. The estimate of
the unknown
and the margin of error is

Creating Confidence Intervals
Parameter – Identify the population of interest
and the parameter you want to draw conclusions
about.
2. Conditions – Choose the appropriate inference
procedure. Verify conditions for using it.
3. Calculations – If the conditions are met, carry out
the inference procedure.
1.
4.
Interpretation – Interpret your results in the
context of the problem. Remember the “three
C’s”: conclusion, connection, and context.
Constructing a Confidence Interval for

A manufacturer of high-resolution video terminals
must control the tension on the mesh of fine wires
that lie behind the viewing screen to prevent
wrinkles and tearing the screen. The tension is
measured by 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
The tension readings from
an SRS of 20 screens are given below. Construct
and interpret a 90% confidence interval for
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
Constructing a Confidence Interval for
Step 1 – Parameter – The population of interest is
all video terminals produced on the day in
question. We want to estimate the mean
tension for all of these screens.
 Step 2 – Conditions – We are told we have an
SRS. Normality is met because past experience
suggests that the tension readings of screens
produced on a single day follow a Normal
distribution quite closely. What if we didn’t know
the distribution was Normal? Could we use the
CLT? What would we do? We must assume that
at least 200 video terminals were made this day
for independence to be true.

Constructing a Confidence Interval for

Step 3 – Calculation
 We need to know
We can find this easily with our
calculator. We also need to know z* for a confidence
level of 90%. Looking up 0.95 in the table, we find that
z* = 1.645.
 So the 90% Confidence Interval for
is

Step 4 – Interpretation
 We are 90% confident that the true mean tension in the
entire batch of video terminals produced that day is
between 290.5 and 322.1 mV.
Another Confidence Interval

Suppose that a single computer screen had a tension
reading of 306.3 mV, the same value as the mean of the 20
screens in our example. What would the confidence interval
be if we based it on only this single value? (Note: all of the
conditions are the same and we have already shown that we
met them.)

The mean of 20 measurements gives a smaller margin of
error and therefore a shorter interval than a single
measurement. This is shown below.
Let’s Try Some

Find z* for a confidence level of 92.5%. (Note: A
picture may help.)

10.9 on p. 6.32
How Confidence Intervals Behave
The user chooses a confidence level, and the
margin of error follows from this choice. Ideally,
we would like a high confidence and a small
margin of error.
 High confidence says that our method almost
always gives correct answers and a small margin
of error says that we have pinned the parameter
down quite precisely.

Smaller Margin of Error

Since
gets smaller when:
the margin of error
 z* gets smaller. As z* gets smaller, so does our
confidence. So to obtain a smaller margin of error, we
must be willing to accept lower confidence.

gets smaller. The standard deviation measures the
variation in the population. It is very difficult for us to
make smaller.
 n gets larger. Increasing the sample size n reduces the
margin of error for any fixed confidence level. Because
we take the square root of n, we must take four times as
many observations to cut the margin of error in half.
Determining Sample Size
A wise user of statistics never plans data
collection without also planning for inference at the
same time. We can arrange to have both a high
confidence and a small margin of error by taking
enough observations.
 The margin of error m of the confidence interval
for the population mean
To
determine the sample size for a desired margin of
error m, substitute the value of z* for your desired
confidence level, set the expression for m less
than or equal to the specified margin of error, and
solve the inequality for n. Let’s look at an
example.

Determining the Sample Size
Researchers would like to estimate the mean
cholesterol level of a particular type of monkey
within 1 milligram per deciliter of blood (mg/dl) of
the true value at a 95% confidence level. The
standard deviation is about
Since
monkeys are expensive, they want to find the
minimum number monkeys they need to control
cost.
 At a 95% confidence level, z* = 1.96. So

Determining Sample Size
When you find the sample size, you always round
up because it has to be what you got or bigger
and rounding down would be smaller.
 Here is the principle:


Notice that it is the size of the sample that
determines the margin of error. The size of the
population does not influence the sample size we
need.
Cautions/Reminders

To use this method of creating a confidence
interval:




Data must be from an SRS of the population.
Different methods are needed if it is not an SRS.
There is no correct method for inference from bad data.
Outliers can distort the results because of their impact
on the mean.
 The shape of the population distribution matters.
 You must know the standard deviation of the population
to do this. We will learn how to handle this without the
standard deviation of the population in the next section.
 95% is the confidence we have in how often the
procedure gives correct results not the probability that
the true mean falls within our interval.
10.20
Estimating a Population Mean
HW: 10.27, 10.28, 10.30, 10.32, 10.34,
10.35, 10.36, 10.40, 10.42

These conditions are very important and must
always be checked before doing any inference.
If Conditions are Met
We know that the sample mean has the Normal
distribution with mean and standard deviation
 Because we don’t know
we estimate it by the
sample standard deviation s. We then estimate
the standard deviation of by
We call this
quantity the standard error of the sample mean

The t Distributions
When we know the value of we base a
confidence interval for on the sampling
distribution of
which has a Normal distribution if
the conditions are satisfied.
 When we do not know
we substitute the
standard error
of for its standard deviation
 The distribution of the resulting statistic, t, is not
Normal. It is a t distribution.
 Unlike the Normal distribution, there is a different t
distribution for each sample of size n. We specify
a particular t distribution by giving its degrees of
freedom (df).

The t Distributions (cont.)

When we perform inference about using a t
distribution, the appropriate degrees of freedom is
We will write the t distribution with k degrees of
freedom as
for short.
 We will also refer to the standard Normal
distribution as the z distribution.

Facts about t Distributions
The density curves of t distributions are similar in
shape to the standard Normal curve. They are
symmetric about zero, single-peaked, and bellshaped.
 The spread (variation) of the t distribution is a bit
greater than that of the standard Normal
distribution. This is true because substituting the
estimate s for the parameter introduces more
variation.
 As the degrees of freedom increase, the
density curve approaches the z distribution even
more closely. This happens because s does a
better job of estimating with a large sample.

Using a t Distribution
Table C in the back of the book gives critical
values t* for the t distribution. Each row contains
critical values for one of the t distributions with the
degrees of freedom listed at the left of the row.
 Several of the more common confidence levels C
are given at the bottom of the table.
 Technology makes use of such tables
unnecessary, but we still need to know how to use
them.

Finding t*
Suppose we want to construct a 95% confidence
interval for the mean of a population based on
an SRS of size
What critical value t*
should we use?
 Just look in table C for a confidence level of 95%
and df = 11.
 So our critical value is
2.201.

One-Sample t Confidence Intervals

To construct a confidence interval for based on
a sample from a Normal population with an
unknown
replace the standard deviation
of
by its standard error
in the formula from
section 1. Use critical values from the t
distribution with
degrees of freedom in place
of the z critical values. This interval is calculated
the same as the one in section 1 with these minor
changes.
Constructing a One-sample Interval

Environmentalists, government officials, and
vehicle manufacturers are all interested in
studying the auto exhaust emissions produced by
motor vehicles. The major pollutants in auto
exhaust from gasoline engines are hydrocarbons,
monoxide, and nitrogen oxides (NOX). The table
below gives NOX levels (in grams per mile) for a
random sample of light-duty engines of the same
type.
1.28
1.17
1.16
1.08
0.60
1.32
1.24
0.71
0.49
1.38
1.20
0.72
0.95
2.20
1.78
1.83
1.26
1.73
1.31
1.80
1.15
0.97
1.12
1.79
1.31
1.45
1.22
1.32
1.47
1.44
0.51
1.49
1.33
0.86
0.57
2.27
1.87
2.94
1.16
1.45
1.51
1.47
1.06
2.01
1.39
0.78
Constructing a One-sample Interval
Step 1 – Parameter – The population of interest is
all light-duty engines of this type. We want to
estimate the mean amount of the pollutant NOX
emitted, for all of these engines.
 Step 2 – Conditions – The data come from a
“random sample” of 46 engines from the
population. We are not told that it is an SRS. Our
calculations may be slightly off if a different
sampling method was used. The problem does not
tell us whether the population is Normal. We will
check this condition on the next slide. We must
assume that at least 460 of these types of engines
have been made for independence to be true.

Constructing a One-sample Interval
To check this, we need to plot the data to see if it
is symmetric and approximately normal. We also
need to check to see if it fits the 68-95-99.7 rule.
 The TI’s boxplot shows that the
data are approximately
symmetric, if you ignore the
outliers. The histogram also
looks approximately Normal if
you again ignore the outliers. The table below
also shows that we are very close to the 68-95Within __ Standard Deviations
1
2
3
99.7 rule. We
Count
33
45
46
will assume
Percent
71.7% 97.8% 100%
Normality.

Constructing a Confidence Interval for

Step 3 – Calculation
 We need to know
We can find this
easily with our calculator. We also
need to know t* for a confidence level
of 95%. Looking up 95% confidence in
the t table with df = 45, we find that
t* = 2.021. Why did we use 40 and not 50?
 So the 95% Confidence Interval for
is

Step 4 – Interpretation
 We are 95% confident that the true mean level of NOX
emitted by this type of light-duty engine is between
1.185 and 1.473 grams/mile.
What if df is not in the Table
When the actual degrees of freedom does not
appear in the table, we use the largest degrees of
freedom in the table that is smaller than our
desired degrees of freedom. This allows us to
overestimate by creating a wider confidence
interval than we need. So we are being safe.
 Of course, you could just use your calculator!
 If we go to distribution, we can use invT(0.975, 45)
to get the value of t*.

Paired t Procedures
Comparative studies are more convincing than
single-sample investigations. For that reason, one
sample inference is less common than
comparative inference.
 A common design to compare two treatments
makes use of one-sample procedures. Recall
from Chapter 5 that is a matched pairs design,
subjects are matched in pairs and each treatment
is given to one subject in each pair. Alternatively,
each subject receives both treatments in some
order that is randomly selected.
 Another situation calling for paired t procedures is
before-and-after observation on the same
subjects.


The parameter
in a paired t procedure is
 the mean difference in the responses to the two
treatments within matched pairs of subjects in the entire
population (when subjects are matched in pairs), or
 the mean difference in response to the two treatments
for individuals in the population (when the same subject
receives both treatments), or
 the mean difference between before-and-after
measurements for all individuals in the population (for
before-and-after observations on the same individuals).
Example of Paired t Procedure

Is caffeine dependence real? Our subjects are 11
people who have been diagnosed as being
caffeine dependent. Each subject was barred
from coffee, colas, and other substances
containing caffeine. Instead they took capsules
containing their normal caffeine intake. During a
different time period, they took placebo capsules.
The order in which they took caffeine and placebo
was randomized. Depression is the score on the
Beck Depression Inventory. Higher scores show
more symptoms of depression. Let’s create a 90%
confidence interval for the mean change in
depression score. Scores are on the next slide.

Subject
Depression
(caffeine)
Depression
(placebo)
Difference
(placebo – caffeine)
1
5
16
11
2
5
23
18
3
4
5
1
4
3
7
4
5
8
14
6
6
5
24
19
7
0
6
6
8
0
3
3
9
2
15
13
10
11
12
1
11
1
0
-1
Step 1 – Parameter – The population of interest is all
people who are dependent on caffeine. We want to
estimate the mean difference
in depression score that would be reported if all
individuals in the population took both the caffeine and
placebo capsules.
Example of Paired t Procedure

Step 2 – Conditions
 SRS – These individuals may not be an SRS. Such
groups are usually volunteers. So they may not be truly
representative of the population. As a result, we may
have trouble generalizing the results of this study to the
population. However, since the researchers randomly
assigned the order in which each subject took the
placebo and the caffeine tablet, any consistent
differences we observe in subjects’ responses should
be due to the treatments.
 Normality – We don’t know so we have to check by
looking at a boxplot and seeing if it matches the 68-95Within __ Standard Deviations
1
2
3
99.7 rule.
Count
8
11
11
Both check.
Percent
72.7%
100%
100%
Example of Paired t Procedure
 Independence – Obviously one individuals depression level
should be independent from another individuals.

Step 3 – Calculation
 We need to know
We can find this easily with our
calculator. We also need to know t* for a confidence level of
90%. Looking up 90% confidence in the t table with df = 10,
we find that t* = 1.812.
 So the 90% Confidence Interval for
is

Step 4 – Interpretation
 We are 90% confident that the mean difference in
depression score for the population is between 3.584 and
11.144 points. So on average, we estimate that caffeinedependent individuals would score between 3.584 and
11.144 points higher on the Beck Depression Inventory when
they are given a placebo instead of caffeine.
Example of Paired t Procedure
 This study provides evidence that withholding caffeine
from caffeine-dependent individuals may lead to
depression. Since the subjects were not an SRS,
however, we cannot generalize any further.
Many studies that use a Paired t Procedure
involve individuals who are not from an SRS of the
population. In such cases, we may not be able to
generalize our findings to the population of
interest.
 By randomizing the treatments, however, we can
ensure that the sizable differences in mean scores
is attributable to the treatment.

Randomness

Random selection of individuals – allows us to
generalize to the entire population from which the
individuals were selected

Random assignment of treatments – allows us to
investigate whether there is a treatment effect,
which might suggest that the treatment caused the
observed difference.

Be sure you understand the difference between
these two!
Robustness of t Procedures

The t confidence interval is exactly correct when
the distribution of the population is exactly Normal.
No real data are exactly Normal. The usefulness
of the t procedures in practice therefore depends
on how strongly they are affected by lack of
Normality. Procedures that are not strongly
affected are called robust.

The t procedures are not robust against outliers,
because
are not resistant to outliers.
NOX Emissions

Earlier we constructed a confidence interval for
the mean level of NOX emissions by a specific
type of light-duty engine. One of the recorded
values in our sample was an extremely high outlier
(2.94 grams/mile). If we remove this data point,
the remaining 45 values have
The confidence interval based on this sample of
45 values would be (1.165, 1.421). This new
confidence interval is narrower and is centered at
a lower value that the original interval (1.185,
1.473). Are either of these correct? The outlier
suggests that the population distribution of NOX
emissions may not be normal and therefore, the
results would not be correct.
Robustness of t Procedures
Fortunately, t procedures are quite robust against
non-Normality of the population when there are no
outliers, especially when the distribution is roughly
symmetric. Larger samples improve the accuracy
of critical values from the t distribution when the
population is not Normal because of the CLT.
 Always make a plot to check for skewness and
outliers before you use the t procedures for small
samples. We can generally safely use the onesample t procedure when
unless an outlier
or quite strong skewness is present.

If your sample data would produce biased data for
some reason, then you shouldn’t bother
computing a t interval.
 If the data you have is the entire population of
interest, then there’s no need to perform inference.

Can We use the t Distribution?

Percent of residents aged 65 and over in the
states.

No; this is a population, not a sample.
Can We use the t Distribution?

Times of the first lightning strike each day at a site
in Colorado

Yes; It is a sample with over 30 observations that
is approximately symmetric.
Can We use the t Distribution?

Word lengths in Shakespeare’s plays

Yes; if the sample size is large enough to
overcome the population’s right-skewness.
Estimating a Population Proportion
HW: 10.46, 10.47, 10.48, 10.50, 10.52,
10.54, 10.56, 10.58
Population Proportions
So far we have discussed making inferences
about population means. We often want to know
percentages for a population though.
 For example:

 What proportion of U.S. adults are unemployed right




now?
What proportion of teenagers have a computer with
internet access in their bedroom?
What proportion of college students pray on a daily
basis?
What proportion of preteens have a cell phone?
What proportion of Californians approve of the
President’s handling of the situation in Iraq?
Binge Drinking in College

Alcohol abuse is obviously a problem for colleges
and universities. We want to know how prevalent
binge drinking is. A 2001 survey of 10,904 U.S.
college students collected information on drinking
behavior and alcohol-related problems. The
researchers defined “frequent binge drinking” as
having five or more drinks in a row three or more
times in the past two weeks. According to this
definition, 2,486 students were classified as binge
drinkers. What proportion of students in this
sample were binge drinkers? Based on this data,
what can we say about the proportion of all
college students who engage in binge drinking?
Estimating a Population Proportion
We want to know what the population proportion p
for a given population is. To do this, we use the
statistic
 In the previous example,
So we
believe that the actual proportion of U.S. college
students who binge drink (drink 5 or more drinks in
a row at least three times over the past two
weeks) is around 23%.
 Notice that I have to define what I mean. For
instance, did you know that a survey found that
20% of adolescents smoke.
 Actually, 20% had smoked at least once in the
past month. Only 4% had smoked at least half a
pack on at least 20 of 30 days in the month.

Sampling Distribution of a Sample Proportion

Important properties of the sampling distribution of
the sample proportion
 Center – The mean is p. In other words, the sample
proportion is an unbiased estimator of the population
proportion p.
 Spread – The standard deviation of
is
provided that the population is at least
10 times as large as the sample.
 Shape – If the sample size is large enough that both np
and n(1-p) are at least 10, the distribution of is
approximately Normal.
But we are Trying to find p
Obviously, we don’t know the population
proportion p. If we did, we wouldn’t have to create
a confidence interval.
 For this reason, we cannot check to see if np and
n(1-p) are at least 10.
 In large samples, should be close to p.
Therefore, we replace p with to determine the
values of np and n(1-p).
 We also replace the standard deviation with the
standard error of

Conditions for Inference about a Proportion

If you have a small sample or a sampling design
more complex than an SRS, you can still do
inference, but you have to use other methods. We
will not be discussing those methods.
More on Binge Drinking

We want to use our data to give a confidence
interval for the proportion of college students who
have engaged in frequent binge drinking. Can
we?
 SRS – It is not an SRS. It is actually a complex
stratified random sample. The overall effect is close to
an SRS though.
 Normality – The counts of “Yes” and “No” are both
greater than 10:
 Independence – Since there are over 109,040 college
students in the U.S., the population is more than 10
times the size of the sample.

Note: The confidence interval is still of the form
Confidence Interval for Binge Drinking
We want to create a 99% confidence interval for
the proportion of U.S college students who
frequently binge drink.
 What we know:


So the confidence interval is

We are 99% confident that the proportion of U.S.
college students who engage in frequent binge
drinking is between 0.218 and 0.238.
Remember
The margin of error in this confidence interval
includes only random sampling error.
 The other sources of error that are not taken into
account – respondents lying, nonresponse rate,
not having an SRS – result in bias.

Teens Say Sex Can Wait
The 2004 Gallup Youth Survey asked a random
sample of 439 U.S. teens aged 13 to 17 whether
they thought young people should wait to have
sex until marriage. 246 said “Yes.” Construct a
95% confidence interval for the proportion of teens
who would say “Yes” if asked this question.
 Step 1 – Parameter – The population is all U.S.
teens. The parameter of interest is the proportion
of all U.S. teens aged 13 to 17 who would say
“Yes” young people should wait to have sex until
marriage.

Teens Say Sex Can Wait

Step 2 – Conditions
 SRS – It is not actually an SRS, but they were chosen
using a method that is designed to ensure a
representative sample (which is why we need an SRS).
 Normality – It is approximately normal since
 Independence – Since there are more than 4,390 teens
aged 13 to 17 in the U.S., the population is more than
10 times larger than our sample size.
Teens Say Sex Can Wait

Step 3 – Calculations

Step 4 – Conclusions – We are 95% confident that
the true population proportion of U.S teens aged
13 to 17 who would say “Yes” teens should wait
until they are married to have sex would be within
the interval (0.514, 0.606). This depends on
whether our sample is truly representative.
Choosing the Sample Size
In a planning a study, we may want to choose a
sample size that will allow us to estimate the
population proportion within a given margin of
error. We did this same thing for a population
mean and the procedure is similar for a population
proportion.
 The margin of error in the approximate confidence
interval for p is

Choosing the Sample Size

Because the margin of error uses the sample
proportion of successes we need to guess this
value when choosing n. Call our guess p*. There
are two ways to get p*.
1. Use a guess for p* based on a pilot study or on past
experience with similar studies. You should do several
calculations that the cover the range of
you
might get.
2. Use p* = 0.5 as the guess. The margin of error is
largest when
so this guess is conservative in
the sense that if we get any other when we do our
study, we will get a margin of error smaller than
planned.

Which method for finding the guess p* should we use.
The n we get doesn’t change much when you change
p* as long as p* is not too far from 0.5. So use the
conservative guess p* = 0.5 if you expect the true p
to be roughly between 0.3 and 0.7. If the true p will
be closer to 0 or 1, using p* will give a sample larger
than needed. So try to use a guess from a pilot study
when you suspect p will be less than 0.3 or greater
than 0.7.
Customer Satisfaction
A company received complaints about its
customer service. They want to conduct a survey
to find out whether this is a true problem or just a
few incidents. They want to find the proportion of
customers that are “satisfied” or “very satisfied”
within 3% at a 95% confidence level. Since we
have no idea what the actual proportion p is, we
decide to be safe and use p* = 0.5.
 So
which means we need a
sample size of at least 1068 to
ensure that the margin of error
is no more than 3%.

Let’s Try a Few!

10.55, 10.57, 10.59