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AP Statistics Chapter 10
Introduction to Inference
Confidence Intervals:
The Basics
Objectives:
-Interpret confidence levels and confidence intervals
-Construct a confidence interval
-Use confidence intervals wisely
.
Point Estimator and
Point Estimate
A point estimator is a statistic that provides an estimate of a population parameter.
The value of that statistic from a sample is called the point estimate. Ideally, a point
estimate is our “best guess” at the value of an unknown parameter
Confidence Interval,
Margin of Error,
Confidence Level
A confidence interval for a parameter has two parts:
 An interval calculated from the data, which has the form
estimate  margin of error
or
statistic  (critical value)* (standard deviation of statistic)
The margin of error tells how close the estimate tends to be to the unknown
parameter in repeated random sampling.

A confidence level C, which gives the overall success rate of the method for
calculating the confidence interval. That is, in C% of all possible samples, the
method would yield an interval that captures the true parameter value.
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.
Interpreting Confidence
Levels 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].”
The price we pay for greater confidence is a wider interval. If we’re satisfied with
80% confidence, then our interval of plausible values for the parameter will be much
narrower than if we insist on 90%, 95%, or 99% confidence.
AP Exam Tip
On a given problem, you may be asked to interpret the confidence interval, the confidence
level, or both. Be sure you understand the difference: the confidence level describes the longrun capture rate of the method, and the confidence interval gives a set of plausible values for
the parameter.
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Calculating a Confidence
Interval
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.
Facts about CI


The confidence level decreases. There is a trade-off between the confidence
level and the margin of error. To obtain a smaller margin of error from the
same data, you must be willing to accept lower confidence.
The sample size n increases. We like high confidence and small margin of
error. A small margin of error says that we have pinned down the parameter
quite precisely. (Remember that  pˆ 
p(1  p)

and  x 
. So as
n
n
the sample size n increases, the standard deviation of the statistic decreases.)

Conditions for
Constructing a CI


Random: The data come from a well-designed random sample or
randomized experiment
Normal: The sampling distribution of the statistic is approximately Normal
Independent: Individual observations are independent.
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8.2 Confidence
Intervals: Estimating a
Population Proportion
Objective:
-Carry out the steps in constructing a confidence interval for a population proportion: define
the parameter; check conditions; perform calculations; interpret results in context.
-Determine the sample size required to obtain a level C confidence interval for a population
proportion with a specified margin of error.
-Understand how the margin of error of a confidence interval changes with the sample size
and the level of confidence C.
- choose a sample size
Assignment 8.2 page 496 to #27-47 odd, 49-54 all
Standard Error
When the standard deviation of a statistic is estimated from data, the result is
called the standard error of the statistic.
The sample proportion p̂ is the statistic we use to estimate p. When the
Independent condition is met, the standard deviation of the sampling
distribution of p̂ is
p(1  p)
n
 pˆ 
If we don’t know the value of p, we replace it with the sample proportion p̂
pˆ (1  pˆ )
n
This quantity is called the standard error (SE) of the sample proportion p̂ . It
describes how close the sample proportion p̂ will be, on average, to the
population proportion p in repeated SRSs of size n.
Construct a Confidence
Interval for p
One-Sample z Interval for a Population Proportion:
Choose an SRS of size n from a large population with unknown p of successes. A
level C confidence interval for p is
Formula: Estimate  margin of error
Statistic  (critical value)*(Standard deviation of statistic)
pˆ  z 
pˆ (1  pˆ )
n
Calculator: STAT  TESTS 1-PropZ Interval  Data or Stats
C.I.
80%
90%
95%
99%
Tail area
0.1
0.05
0.025
0.005
z*(Critical Value)
1.282
1.645
1.960
2.576
Conditions for constructing a CI for p:
1. Data must come from a random sample or randomized experiment
2. At least 10 successes and failures; that is npˆ  10 and n(1  pˆ )  10
3. Observations are independent; 10% condition is met
Confidence Intervals:
Four-Step Process
State: What parameter do you want to estimate, and at what confidence level?
Plan: Identify the appropriate inference method. Check conditions.
Do: If the conditions are met, perform the calculations.
Conclusion: Interpret your interval in the context of the problem
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Choosing a Sample Size
Choosing a Sample Size for Desired Margin of Error
To determine the sample size n that will yield a level C confidence interval for a
population proportion p with a maximum margin of error ME, solve the following
inequality for n:
pˆ (1  pˆ )
 ME
n
where p̂ is a guessed value for the sample proportion. The margin of error will
always be less than or equal to ME if you take the guess p̂ to be 0.5
z*
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8.3 Confidence
Intervals: Estimating a
Population Mean
1. Read Section 8.3 along
with these notes
2. Do Assignment 8.3 Page
518 #55-73 odd, 75-80
DUE TOMORROW!!!!
(Good Luck! I Believe in You)
The z distribution
Objective:
- When  is known, construct and interpret a one-sample z interval for a population mean
- When  is unknown, construct and interpret a one-sample t interval for a population mean
- -Determine the sample size required to obtain a level C confidence interval for a population
mean with a specified margin of error.
One-Sample z Interval for a Population Mean:
Choose an SRS of size n from a population with unknown mean  and known
standard deviation  of successes. A level C confidence interval for  is
Formula:
x  z

n
Calculator: STAT  TESTS Z Interval  Data or Stats
Conditions for constructing a CI for  :
1. Random: Data must come from a random sample or randomized experiment
2. Normal: Population distribution is normal or a large sample ( n  30 )
3. Independent: Observations are independent; 10% condition is met
Choosing a Sample Size
Choosing a Sample Size for Desired Margin of Error When Estimating 
To determine the sample size n that will yield a level C confidence interval for a
population mean  with a maximum margin of error ME, solve the following
inequality for n:
z*

n
 ME
(Always roundup when finding n)
The t distribution When we do not know σ, we substitute the standard error (SE)

sx
n
of x for its s.d.
n . The statistic that results does not have a normal dist. It has a distr. called
the t distribution.
The t distribution has a different shape than the standard Normal curve: still
symmetric with a single peak at 0, but with much more area in the tails.
The statistic t has the same interpretation as any standardized statistic: it says how far
x is from its  in standard deviation units. There is a different t distribution for each
sample size. We specify a particular t distribution by giving its degrees of freedom
(df).
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The t Distribution; Degrees of Freedom
Choose an SRS of size from a large population with mean  and standard deviation

t
Facts about the t distribution:



x
sx
with degrees of freedom df = n-1.
n
The density curves of the t distr. are similar in shape to the standard normal
curve. They are symmetric about zero, single peaked, and bell shaped.
The spread of the t distribution is a bit greater than the standard normal distr.
The t distribution has more probability in the tails and less in the center than
the normal.
As the degrees of freedom increase, the t density curve approaches the N(0,1)
curve. This happens because s x for the fixed parameter 
causes little extra variation when the sample is large.
One-Sample t Interval for a Population Mean:
Choose
an
SRS
of
size n from a population with unknown mean. A level C
t Confidence Interval
confidence interval for  is
Formula:
x t
sx
n
with df = n – 1
Where t* is the critical value for t n 1 distribution.
Table B gives critical values for the t distribution. By looking down any column, you
can check that the t critical values approach the normal values as the degrees of
freedom increase. When the actual df does not appear in the table, use the greatest df
available that is less than the greatest df available.
Calculator: STAT  TESTS T Interval  Data or Stats
Conditions:
1. Random: Data must come from a random sample or randomized experiment
2. Normal: Population distribution is normal or a large sample ( n  30 )
3. Independent: Observations are independent; 10% condition is met
An inference procedure is called robust if the probability calculations involved in that
Robust procedures procedure remain fairly accurate when a condition for using the procedure is violated.
The t procedures are not robust against outliers, because x and s x are not resistant
to outliers.




Using One sample t Procedures
Except in the case of small samples, the assumption that the data are an SRS
from the population of interest is more important than the assumption that the
population distribution is normal.
Sample size less than 15. Use t procedure if the data are 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.
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