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Chapter 8: Estimating with Confidence
Section 8.1
Confidence Intervals: The Basics
The Practice of Statistics, 4th edition – For AP*
STARNES, YATES, MOORE
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Chapter 8
Estimating with Confidence
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8.1 Confidence Intervals: The Basics
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8.2 Estimating a Population Proportion
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8.3 Estimating a Population Mean
+ Section 8.1
Confidence Intervals: The Basics
Learning Objectives
After this section, you should be able to…
✓
INTERPRET a confidence level
✓
INTERPRET a confidence interval in context
✓
DESCRIBE how a confidence interval gives a range of plausible
values for the parameter
✓
DESCRIBE the inference conditions necessary to construct
confidence intervals
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EXPLAIN practical issues that can affect the interpretation of a
confidence interval
sample mean X̄ is likely to be. All confidence intervals we
construct will have a form similar to this:
Definition:
A confidence intervals for a parameter has two parts:
An interval calculated from the data, which has the form:
Estimate（估计值） ± margin of error （边际误差）
The margin of error tells how close the estimate tends to be to
the unknown parameter in repeated random sampling.
Confidence Intervals: The Basics
The sampling distribution of X̄ tells us how close to u the
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The idea of condifidence level
题⽬目应⽤用：已知区间（a, b），求estimate和margin of error
给定置信区间（0.201，0.876），求这个置信区间的estimator 和
margin of error
Estimate:
Margin of error:
Confidence Intervals: The Basics
Estimate ± margin of error （边际误差）
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The idea of condifidence level
Estimate ± margin of error
u= X̄ ±10
In repeated samples, the value of
X̄ follow a normal distribution with
mean and standard deviation 5
The 66-95-99 rule tell us that in
95% of these samples of size 16,
X̄ will be within 10(two standard
deviations) of u.
If X̄ is within 10 points of u, then
u is within 10 points X̄
Confidence Intervals: The Basics
To estimate the population mean u, we can use X̄ =240.79 as a point
estimate, we don't expect u to be exactly equal to X̄ , so we need to
say how close to 240.79 is u likely to be?
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The Idea of a Confidence Interval
Margin of error取决于我们⾃自⼰己希望这个区间包含总体参数的结果具有多⼤大
的可信程度（概率）
Probably not. However, u is probably somewhere around 240.79.
How close to 240.79 is u likely to be ?
(to answer this question, we must answer another)
How would the sample mean X̄ vary if we took many SRSs?
Confidence Intervals: The Basics
Consider: is the value of the population mean u exactly 240.79?
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The idea of condifidence level-how to
construct the confidence interval
Shape: since the population is normal, so is the sampling distribution
of X̄
Center: the mean of the sampling distribution of X̄ equals to u, the
mean of the population distribution.
Spread: the standard deviation of the sampling distribution of X̄ for
an SRSs of n=16 observations is (if σ=20)?
Confidence Intervals: The Basics
How would the sample mean X̄ vary if we took many SRSs?
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The idea of condifidence level
信⼼心程度：95%
精确度：Estimate ± margin of error 区间⻓长度
⼆二者对⽴立
降低confidence level 68%
Confidence Intervals: The Basics
Margin of error取决于我们⾃自⼰己希望这个区间包含总体参数的结果具有
多⼤大的可信程度（概率）
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The Idea of a Confidence Interval
Interpreting Confidence Level and Confidence Intervals
Confidence level: To say that we are 95% confident is
shorthand for “95% of all possible samples of a given
size from this population will result in an interval that
captures the unknown parameter.”
Confidence interval: To interpret a C% confidence
interval for an unknown parameter, say, “We are C%
confident that the interval from _____ to _____ captures
the actual value of the [population parameter in
context].”
Confidence Intervals: The Basics
The confidence level is the overall capture rate if the method is used many
times. Starting with the population, imagine taking many SRSs of 16
observations. The sample mean will vary from sample to sample, but when we
use the method estimate ± margin of error to get an interval based on each
sample, 95% of these intervals capture the unknown population mean µ.
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Interpreting Confidence Levels and Confidence Intervals
2010 FRQ 03
3. A human society wanted to estimate with 95 percent confidence the
proportion of households in its country that own at least one dog
(1)interpret the 95 percent confidence
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考点练习：
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考点练习：
2010 FRQ 03
3. A human society wanted to estimate with 95 percent confidence the
proportion of households in its country that own at least one dog
(1)interpret the 95 percent confidence
Solution:
2010 FRQ 03
The 95 percent confidence level means that if one were to repeatedly take
random samples of the same size from the population and construct a 95
percent interval from each sample, then in the long run 95 percent of those
intervals would succeed in capturing the actual value of the population
proportion of households in the country than own at least on dog.
(A) In this population, about 95 percent of all rental prices are between
$815 and $825.
(B) In this sample, about 95 percent of the 100 rental prices are between
$815 and $825.
(C) In repeated sampling, the method produces intervals that include the
population mean approximately 95 percent of the time.
(D) In repeated sampling, the method produces intervals that include the
sample mean approximately 95 percent of the time.
(E) There is a probability of 0.95 that the true mean is between $815 and
$825.
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考点练习：
Monthly rent was determined for each apartment in a random sample
of 100 apartments. The sample mean was $820 and the sample
standard deviation was $25. An approximate 95 percent confidence
interval for the true mean monthly rent for the population of apartments
from which this sample was selected is ($815, $825). Which of the
following statements is a correct interpretation of the 95 percent
confidence level?
(A) Approximately 95 percent of alfalfa fields sampled will have an average of 1.50
to 3.50 alfalfa weevils per plant, but nothing can be said about the sample mean
number of alfalfa weevils for this or any other field.
(B) Approximately 95 percent of alfalfa fields sampled will have an average of 1.50
to 3.50 alfalfa weevils per plant. The sample mean for this field was 2.50 alfalfa
weevils per plant, but the sample means for other fields may be different.
(C) If we repeatedly sampled this field, taking samples of 80 plants and constructing
95% confidence intervals, then, approximately 95 percent of these intervals would
include 2.5, the mean for the sample described above.
(D) If we repeatedly sampled this field, taking samples of 80 plants and constructing
95% confidence intervals, then, approximately 95 percent of these intervals would
include the population mean number of alfalfa weevils on an alfalfa plant in this field.
(E) If we repeatedly sampled this field, taking samples of 80 plants and constructing
95% confidence intervals, then, approximately 95 percent of these intervals would
include the sample mean for that sample.
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A researcher constructed a 95 percent confidence interval for the mean number of
alfalfa weevils on an alfalfa plant within a field. Based on 80 randomly selected
alfalfa plants, the researcher found an average of 2.5 alfalfa weevils per plant and
computed the 95 percent confidence interval to be 1.50 to 3.50. Which of the
following statements is a correct interpretation of the 95 percent confidence level?
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A polling agency conducted a survey by selecting 100 random samples,
each consisting of 1,200 United States citizens. The citizens in each
sample were asked whether they were optimistic about the economy. For
each sample, the polling agency created a 95 percent confidence interval
for the proportion of all United States citizens who were optimistic about the
economy. Which of the following statements is the best interpretation of the
95 percent confidence level?
(A) With 100 confidence intervals, we can be 95% confident that the
sample proportion of citizens of the United States who are optimistic about
the economy is correct.
(B) We would expect about 95 of the 100 confidence intervals to contain
the proportion of all citizens of the United States who are optimistic about
the economy.
(C) We would expect about 5 of the 100 confidence intervals to riot contain
the sample proportion of citizens of the United States who are optimistic
about the economy.
(D) Of the 100 confidence intervals, 95 of the intervals will be identical
because they were constructed from samples of the same size of 1,200.
(E) The probability is 0.95 that 100 confidence intervals will yield the same
information about the sample proportion of citizens of the United States
who are optimistic about the economy.
Confidence interval: To interpret a C% confidence interval for an
unknown parameter, say, “We are C% confident that the interval from
_____ to _____ captures the actual value of the [population parameter in
context.”
FRQ:
STATE
PLAN
DO
CONCLUDE: we are C%…..
!!!单个区间不不存在概率问题
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Interpreting Confidence Interval
(B) The probability is 0.95 that a randomly selected time for response will
be between 2.8 minutes and 12.3 minutes.
(C) Ninety-five percent of the time the mean time for response is between
2.8 minutes and 12.3 minutes.
(D) We are 95% confident that the mean time for response is between 2.8
minutes and 12.3 minutes.
(E) We are 95% confident that a randomly selected time for response will
be between 2.8 minutes and 12.3 minutes.
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A 95 percent confidence interval for the mean time, in minutes, for a
volunteer fire company to respond to emergency incidents is determined
to be(2.8, 12.3). Which of the following is the best interpretation of the
interval?
(A) Five percent of the time, the time for response is less than 2.8
minutes or greater than 12.3 minutes.
The following command was executed on their calculator:
mean(randNorm(M,20,16))
The result was 240.79. This tells us the
calculator chose an SRS of 16 observations
from a Normal population with mean M and
standard deviation 20. The resulting
sample mean of those 16 values was
240.79.
Your group must determine an interval of reasonable values for the
population mean µ. Use the result above and what you learned about
sampling distributions in the previous chapter.
Share your team’s results with the class.
Confidence Intervals: The Basics
Your teacher has selected a “Mystery Mean” value µ and stored it as
“M” in their calculator. Your task is to work together with 3 or 4
students to estimate this value.
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Activity: The Mystery Mean
A point estimator is a statistic that provides an estimate of a
population parameter. The value of that statistic from a sample is called
a point estimate. Ideally, a point estimate is our “best guess” at the
value of an unknown parameter.
We learned in Chapter 7 that an ideal point estimator will have no bias and
low variability. Since variability is almost always present when calculating
statistics from different samples, we must extend our thinking about
estimating parameters to include an acknowledgement that repeated
sampling could yield different results.
Confidence Intervals: The Basics
Definition:
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Confidence Intervals: The Basics
The confidence level does not tell us the chance that a
particular confidence interval captures the population
parameter.
Instead, the confidence interval gives us a set of plausible values for
the parameter.
We interpret confidence levels and confidence intervals in much the
same way whether we are estimating a population mean, proportion,
or some other parameter.
Confidence Intervals: The Basics
The confidence level tells us how likely it is that the method we
are using will produce an interval that captures the population
parameter if we use it many times.
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Interpreting Confidence Levels and Confidence Intervals
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Confidence level: 1 – α
Sampling Distribution of the Mean
α/2
Intervals
extend from
X − Zα / 2
to
X + Zα / 2
σ
1–α
α/2
x
µx = µ
x1
x2
n
(1-α)100%
of intervals
constructed contain
µ;
(α)100% do not.
σ
n
Confidence Intervals
The critical value is ?
Confidence Intervals: The Basics
For example, to find a 95% confidence interval.
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Critical value
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Confidence Intervals
DCOVA
Confidence
Intervals
Population
Mean
σ Known
Population
Proportion
σ Unknown
#24
Now let’s look at how we can derive the confidence interval:
z = ± (X − µ )/
σ
n
rewriting
σ
= ± (X − µ )
n
σ
For + then z
= (X − µ )
n
σ
or µ = X − z
n
σ
For − then z
= (− X + µ )
n
σ
or µ = X + z
n
σ
σ
X− z
≤µ≤ X+ z
n
n
z
Confidence Interval for µ (σ Known)
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DCOVA
Assumptions
■ Population standard deviation σ is known
■ Population is normally distributed
■ If population is not normal, use large sample (n > 30)
Confidence interval estimate:
where
is the point estimate
Zα/2 is the normal distribution critical value for a probability of
α/2 in each tail
σ/ n is the standard error
X
X ± Z α/2
σ
n
#26
The confidence interval for estimating a population parameter has the form
statistic ± (critical value) • (standard deviation of statistic)
where the statistic we use is the point estimator for the parameter.
Properties of Confidence Intervals:
▪ The “margin of error” is the (critical value) • (standard deviation of statistic)
The margin of error gets smaller, so
the width of margin of error (2ME) is
also gets smaller, when:
✓ The confidence level decreases
✓ The sample size n increases
Confidence Intervals: The Basics
Calculating a Confidence Interval
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Calculating a Confidence Interval
Confidence Intervals: The Basics
A large city newspaper periodically reports the mean cost of
dinner for two people at restaurants in the city. The newspaper
staff will collect data from a random sample of restaurants in the
city and estimate the mean price using a 90 percent confidence
interval. In past years, the standard deviation has always been
very close to $35. Assuming that the population standard
deviation is $35, which of the following is the minimum sample
size needed to obtain a margin of error of no more than $5 ?
(A) 90
(B) 112
(C) 133
(D) 147
(E) 195
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Smallest sample size
Confidence Interval for µ (σ Unknown)
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Assumptions
■ Population standard deviation is unknown
■ Population is normally distributed
■ If population is not normal, use large sample (n > 30)
Use Student’s t Distribution
Confidence Interval Estimate:
X ± tα / 2
DCOVA
S
n
(where tα/2 is the critical value of the t distribution with n -1 degrees
of freedom and an area of α/2 in each tail)
#29
✓
✓
It has a different shape than the standard Normal curve:
It is symmetric with a single peak at 0,
However, it has much more area in the tails.
Estimating a Population Mean
When we standardize based on the sample standard deviation
sx, our statistic has a new distribution called a t distribution.
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When σ is Unknown: The t Distributions
Like any standardize statistic, t tells us how far X̄ is from its mean u in standard
deviation units.
However, there is a different t distribution for each sample size, specified by its
degrees of freedom (df).
The t Distributions; Degrees of Freedom
Draw an SRS of size n from a large population that has a Normal distribution
with mean µ and standard deviation σ. The statistic
has the t distribution with degrees of freedom df = n – 1. The statistic will
have approximately a tn – 1 distribution as long as the sampling
distribution is close to Normal.
Estimating a Population Mean
When we perform inference about a population mean µ using a t
distribution, the appropriate degrees of freedom are found by
subtracting 1 from the sample size n, making df = n - 1. We will
write the t distribution with n - 1 degrees of freedom as tn-1.
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The t Distributions; Degrees of Freedom
✓The density curves of the t distributions
are similar in shape to the standard Normal
curve.
✓The spread of the t distributions is a bit
greater than that of the standard Normal
distribution.
✓The t distributions have more probability in
the tails and less in the center than does the
standard Normal.
✓As the degrees of freedom increase, the t
density curve approaches the standard
Normal curve ever more closely.
We can use Table B in the back of the book to determine critical values t* for t
distributions with different degrees of freedom.
Estimating a Population Mean
When comparing the density curves of the standard Normal
distribution and t distributions, several facts are apparent:
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The t Distributions; Degrees of Freedom
Upper-tail probability p
df
.05
.025
.02
10
1.812
2.228 2.359 2.764
11
1.796
2.201 2.328 2.718
12
1.782
2.179 2.303 2.681
z*
1.645
1.960 2.054 2.326
90%
95%
96%
.01
98%
Confidence level C
In Table B, we consult the row
corresponding to df = n – 1 = 11.
We move across that row to the
entry that is directly above 95%
confidence level.
The desired critical value is t * = 2.201.
Estimating a Population Mean
Suppose you want to construct a 95% confidence interval for the
mean µ of a Normal population based on an SRS of size n =
12. What critical t* should you use?
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Using Table B to Find Critical t* Values
Conditions
The One-Sample
for Inference
t Interval
about
for aaPopulation
PopulationMean
Mean
•Random:
Choose
an The
SRSdata
of size
come
n from
fromaapopulation
random sample
havingofunknown
size n from
mean
theµ.
population
A level C
confidence
of
interest orinterval
a randomized
for µ is experiment.
• Normal: The population has a Normal distribution or the sample size is large
(n ≥ 30).
where t* is the critical value for the tn – 1 distribution.
• Independent: The method for calculating a confidence interval assumes that
Useindividual
this interval
only when:are independent. To keep the calculations
observations
accurate
wheniswe
sample
replacement
from(na ≥finite
(1) reasonably
the population
distribution
Normal
orwithout
the sample
size is large
30),
population, we should check the 10% condition: verify that the sample size
• the
at least
10population
times as large
is nopopulation
more thanis1/10
of the
size.as the sample.
Estimating a Population Mean
The one-sample t interval for a population mean is similar in both
reasoning and computational detail to the one-sample z interval for a
population proportion. As before, we have to verify three important
conditions before we estimate a population mean.
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One-Sample t Interval for a Population Mean
PLAN: If the conditions are met, we can use a one-sample t interval to
estimate µ.
Random: We are told that the data come from a random sample of 20
screens from the population of all screens produced that day.
Normal: Since the sample size is small (n < 30), we must check whether it’s
reasonable to believe that the population distribution is Normal. Examine the
distribution of the sample data.
These graphs give no reason to doubt the Normality of the population
Independent: Because we are sampling without replacement, we must check
the 10% condition: we must assume that at least 10(20) = 200 video terminals
were produced this day.
Estimating a Population Mean
Read the Example on page 508. STATE: We want to estimate
the true mean tension µ of all the video terminals
produced this day at a 90% confidence level.
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Example: Video Screen Tension
DO: Using our calculator, we find that the mean and standard deviation of the
20 screens in the sample are:
df
.10
.05
.025
Since n = 20, we use the t distribution with df = 19
to find the critical value.
18
1.130
1.734
2.101
From Table B, we find t* = 1.729.
19
1.328
1.729
2.093
20
1.325
1.725
2.086
90%
95%
96%
Upper-tail probability p
Therefore, the 90% confidence interval for µ is:
Estimating a Population Mean
Read the Example on page 508. We want to estimate the true
mean tension µ of all the video terminals produced this
day at a 90% confidence level.
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Example: Video Screen Tension
Confidence level C
CONCLUDE: We are 90% confident that the interval from 292.32 to 320.32 mV captures the
true mean tension in the entire batch of video terminals produced that day.
Definition:
An inference procedure is called robust if the probability calculations
involved in the procedure remain fairly accurate when a condition for
using the procedures is violated.
Fortunately, the t procedures are quite robust against non-Normality of
the population except when outliers or strong skewness are present.
Larger samples improve the accuracy of critical values from the t
distributions when the population is not Normal.
Estimating a Population Mean
The stated confidence level of a one-sample t interval for µ is
exactly correct when the population distribution is exactly Normal.
No population of real data is exactly Normal. The usefulness of
the t procedures in practice therefore depends on how strongly
they are affected by lack of Normality.
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Using t Procedures Wisely
Using One-Sample t Procedures: The Normal Condition
• Sample size less than 15: Use t procedures if the data appear close to
Normal (roughly symmetric, single peak, no outliers). If the data are clearly
skewed or if outliers are present, do not use t.
• Sample size at least 15: The t procedures can be used except in the
presence of outliers or strong skewness.
• Large samples: The t procedures can be used even for clearly skewed
distributions when the sample is large, roughly n ≥ 30.
Estimating a Population Mean
Except in the case of small samples, the condition that the data
come from a random sample or randomized experiment is more
important than the condition that the population distribution is
Normal. Here are practical guidelines for the Normal condition
when performing inference about a population mean.
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Using t Procedures Wisely
When we calculated a 95% confidence interval for the mystery
mean µ, we started with
estimate ± margin of error
This leads to a more general formula for confidence intervals:
statistic ± (critical value) • (standard deviation of statistic)
Confidence Intervals: The Basics
Why settle for 95% confidence when estimating a parameter?
The price we pay for greater confidence is a wider interval.
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Constructing a Confidence Interval
1) Random: The data should come from a well-designed random
sample or randomized experiment.
2) Normal: The sampling distribution of the statistic is approximately
Normal.
For means: The sampling distribution is exactly Normal if the population
distribution is Normal. When the population distribution is not Normal, then
the central limit theorem tells us the sampling distribution will be
approximately Normal if n is sufficiently large (n ≥ 30).
For proportions: We can use the Normal approximation to the sampling
distribution as long as np ≥ 10 and n(1 – p) ≥ 10.
3) Independent: Individual observations are independent. When
sampling without replacement, the sample size n should be no more
than 10% of the population size N (the 10% condition) to use our
formula for the standard deviation of the statistic.
Confidence Intervals: The Basics
Before calculating a confidence interval for µ or p there are three
important conditions that you should check.
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Using Confidence Intervals
+ Section 8.1
Confidence Intervals: The Basics
Summary
+ Section 8.1
Confidence Intervals: The Basics
Summary
In this section, we learned that…
✓
The confidence level C is the success rate of the method that produces the
interval. If you use 95% confidence intervals often, in the long run 95% of
your intervals will contain the true parameter value. You don’t know whether a
95% confidence interval calculated from a particular set of data actually
captures the true parameter value.
✓
Other things being equal, the margin of error of a confidence interval gets
smaller as the confidence level C decreases and/or the sample size n
increases.
✓
Before you calculate a confidence interval for a population mean or
proportion, be sure to check conditions: Random sampling or random
assignment, Normal sampling distribution, and Independent observations.
✓
The margin of error for a confidence interval includes only chance variation,
not other sources of error like nonresponse and undercoverage.
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Looking Ahead…
In the next Section…
We’ll learn how to estimate a population proportion.
We’ll learn about
✓ Conditions for estimating p
✓ Constructing a confidence interval for p
✓ The four-step process for estimating p
✓ Choosing the sample size for estimating p