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More on Inference
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.
• Twenty-five samples from the same
population provides 25 95% confidence
intervals.
– In the long run, 95% of all samples give an
interval that covers 
CI for population mean
• Choose an SRS of size n from a population
having unknown mean  and known
standard deviation  .
• A level C confidence interval for  is
x  z*

n
Margin of Error
• A small margin of error says that we have pinned
down the parameter quite precisely.

z*
n
• What if the margin of error is too large?
– Use a lower level of confidence
– Increase the sample size
– Reduce 
Choosing the Sample Size
• The confidence interval for a population
mean will have a specified margin of error
m when the sample size is
 z * 
n

 m 
2
Cautions
• Any formula for inference is correct only
in specific circumstances
• The margin of error in a confidence
interval covers only random sampling
errors.
• Review other cautions on page 426
Test Statistic for Hypothesis Testing
• A test statistic measures compatibility between the
null hypothesis and the data.
• It is a random variable with a distribution that we
know.
• When testing the mean with a known variance
(or standard deviation), we use the following test
statistic
x
z 

n
P-value
• The probability, computed assuming that Ho is
true, that the test statistic would take a value as
extreme or more extreme than that actually
observed is called the p-value of the test.
• The smaller the p-value, the stronger the evidence
against Ho provided by the data.
• If the p-value is as small or smaller than alpha, we
say that the data are statistically significant at level
alpha.
CIs and 2-sided Tests
• A level alpha 2-sided significance tests
rejects a hypothesis Ho :   o exactly
when the value  o falls outside a level
1 – alpha confidence interval for
• Fixed alpha tests use the table of standard
normal critical values (Table D)
Use and Abuse
• P-values are more informative than the results of a
fixed level alpha test.
• Beware of placing too much weight on traditional
values of alpha.
• Very small effects can be highly significant,
especially when a test is based on a large sample.
• Lack of significance does not imply that Ho is true,
especially when the test has low power.
• Significance tests are not always valid.
Power
• The probability that a fixed level alpha
significance test will reject Ho when a
particular alternative value of the parameter is
true is called the power of the test to detect that
alternative.
• One way to increase power is to increase
sample size.
– Other suggestions are on page 472.