Transcript Estimation Procedures

Chapter 7
Estimation Procedures
Chapter Outline
 A Summary of the Computation of
Confidence Intervals
 Controlling the Width of Interval
Estimates
 Interpreting Statistics: Predicting the
Election of the President and Judging
His Performance
In This Presentation
 The logic of estimation
 How to construct and interpret
interval estimates for:
 Sample means
 Sample Proportions
Basic Logic
 In estimation procedures, statistics
calculated from random samples are
used to estimate the value of
population parameters.
 Example:
 If we know 42% of a random sample
drawn from a city are Republicans, we
can estimate the percentage of all city
residents who are Republicans.
Basic Logic
 Information from
samples is used to
estimate
information about
the population.
 Statistics are used
to estimate
parameters.
POPULATION
SAMPLE
PARAMETER
STATISTIC
Basic Logic
 Sampling Distribution
is the link between
sample and
population.
 The value of the
parameters are
unknown but
characteristics of the
S.D. are defined by
theorems.
POPULATION
SAMPLING DISTRIBUTION
SAMPLE
Two Estimation Procedures
 A point estimate is a sample statistic
used to estimate a population value.
 A newspaper story reports that 74% of a
sample of randomly selected Americans
support capital punishment.
 Confidence intervals consist of a
range of values.
 ”between 71% and 77% of Americans
approve of capital punishment.”
Constructing Confidence
Intervals For Means
 Set the alpha (probability that the interval
will be wrong).
 Setting alpha equal to 0.05, a 95% confidence
level, means the researcher is willing to be
wrong 5% of the time.
 Find the Z score associated with alpha.
 If alpha is equal to 0.05, we would place half
(0.025) of this probability in the lower tail and
half in the upper tail of the distribution.
 Substitute values into equation 7.2.(text, p.
185)
Confidence Intervals For Means:
Problem 7.5c
 For a random sample of 178
households, average TV viewing was
6 hours/day with s = 3. Alpha = .05.
N=178.




c.i.
c.i.
c.i.
c.i.
=
=
=
=
6.0
6.0
6.0
6.0
±1.96(3/√177)
±1.96(3/13.30)
±1.96(.23)
± .44
Confidence Intervals For Means
 We can estimate that households in this
community average 6.0±.44 hours of TV
watching each day.
 Another way to state the interval:
 5.56≤μ≤6.44
 We estimate that the population mean is greater
than or equal to 5.56 and less than or equal to
6.44.
 This interval has a .05 chance of being
wrong.
Confidence Intervals For Means
 Even if the statistic is as much as
±1.96 standard deviations from the
mean of the sampling distribution the
confidence interval will still include
the value of μ.
 Only rarely (5 times out of 100) will
the interval not include μ.
Other confidence levels (p. 171)
Confidence Alpha
level
90%
.10
Alpha/2
Z score
.05
+/- 1.65
95%
.05
.024
+/1.96
.99
.01
.0050
+/-2.58
99.9%
.001
.0005
+/3.29
Constructing Confidence Intervals
For Proportions
 Procedures:
 Set alpha.
 Find the associated Z score.
 Substitute the sample information into
Formula 7.3. (p. 185)
Confidence Intervals For
Proportions
 If 42% of a random sample of 764 from a
Midwestern city are Republicans, what % of
the entire city are Republicans?
 Don’t forget to change the % to a
proportion.




c.i.
c.i.
c.i.
c.i.
=
=
=
=
.42
.42
.42
.42
±1.96 (√.25/764)
±1.96 (√.00033)
±1.96 (.018)
±.04
Confidence Intervals For
Proportions
 Changing back to %s, we can estimate that
42% ± 4% of city residents are
Republicans.
 Another way to state the interval:
 38%≤Pu≤ 46%
 We estimate the population value is greater than
or equal to 38% and less than or equal to 46%.
 This interval has a .05 chance of being
wrong.
SUMMARY
 In this situation, identify the
following:




Population
Sample
Statistic
Parameter
SUMMARY
 Population = All residents of the
city.
 Sample = the 764 people selected
for the sample and interviewed.
 Statistic = Ps = .42 (or 42%)
 Parameter = unknown. The % of all
residents of the city who are
Republican.