Chapter 6: Introduction to Inference
I. Estimating with Confidence (IPS section 6.1 pages 417-435)
A. Confidence Interval – A level C confidence interval for a parameter is an interval
computed from sample data by a method that has probability C of producing an
interval containing the true value of the parameter.
estimate  margin of error
 The confidence interval shows how confident we are that the procedure will
catch the true population parameter.
 A confidence interval is an interval of the form (a,b), where a and b are
numbers computed from the data.
 It has a property called a confidence level that gives the probability that the
interval covers the parameter.
B. Margin of Error – shows how accurate we believe our guess is, based on the
variability of the estimate. If you calculate a margin of error and decide that it is too
large, the following are your choices to reduce it:
1. Use a lower level of confidence (smaller C)
2. Increase the sample size (larger n)
3. Reduce σ
C. Confidence Interval for a Population Mean – Chose an SRS of size n from a
population having unknown mean μ and known standard deviation σ, A level C
confidence interval for μ is
x  z*

n
Here z* is the value on the standard normal curve with area C between –z* and z*.
This interval is exact when the population distribution is normal and is approximately
correct for large n in other cases.
D. Sample Size for Desired Margin of Error – The confidence interval for a population
mean will have a specified margin of error m when the sample size is
 z * x 
n

 m 
2
Chapter 8: Confidence Intervals for Proportions
II. Inference for a Single Proportion (IPS section 8.1 pages 512-587)
A. Confidence Interval for a Population Proportion
Choose an SRS of size n from a large population with unknown proportion p of
successes. The Wilson estimate of the population proportion is
X 2
p
n4
The standard error of p is
p(1  p)
n4
An approximate level C confidence interval for p is
p  z*SE p
SE p 
Moore, David and McCabe, George. 2002. Introduction to the Practice of Statistics. W. H. Freeman and
Company, New York. 365-413.
where z* is the value for the standard normal density curve with area C between –z*
and z*.
The margin of error is
m  z*SE p
Use this interval when the sample size is at least n = 5 and the confidence level is
90%, 95%, or 99%.
B. Large-Sample Significance Test for a Population Proportion
Draw an SRS of size n from a large population with unknown proportion p of
successes. To test the hypothesis Ho: p = po, compute the z statistic
p  po
z
po (1  po )
n
In terms of a standard normal random variable Z, the approximate P-value for a test of
Ho against
Ha:  >  0 is P(Z  z)
Ha:  <  0 is P(Z  z)
Ha:    0 is 2P(Z  |z|)
C. Sample Size for Desired Margin of Error
The level C confidence interval for a proportion p will have a margin of error
approximately equal to a specified value m when the sample size satisfies
 z*  *
n  4     (1   * )
m
Here z* is the critical value for confidence C, and p* is a guessed value for the
proportion of successes in the future sample. The margin of error will be less than or
equal to m if p* is chosen to be 0.5. The sample size required is then given by
 z* 
n4

 2m 
2
Moore, David and McCabe, George. 2002. Introduction to the Practice of Statistics. W. H. Freeman and
Company, New York. 365-413.