Download Estimation Procedures

Chapter 7
Estimation Procedures
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
Estimation Procedures
 A point estimate is a sample statistic
used to estimate a population value.
 Both sample means and sample
proportions are unbiased estimates of
the population mean or proportion.
(explain bias)
 For both means and proportions we can
use characteristics of their respective
sampling distributions to establish
confidence intervals around the statistic.
Shape of sampling distributions
 For both means and proportions, as N
becomes large, the sampling
distribution will be normal (for our
purposes, N=100 is large enough)
 The standard deviation of the
sampling distribution is called the
standard error.
Standard error and mean of
sampling distribution
1. If we take a large sample, we can
use the mean of the sample as an
unbiased estimate of the mean of the
sampling distribution.
2. The standard error of the mean =
(the standard deviation of the
sample)/(the square root of N-1)
Efficiency of confidence intervals
 Efficiency is determined by the
dispersion in the sampling
distribution.
 The smaller the standard deviation of
the sampling distribution, the greater
the efficiency of our estimate
 Efficiency is therefore maximized as N
gets larger
Constructing Confidence
Intervals For Means
 Set the alpha level(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. Z-scores are
expressed in standard deviation units. Statistics texts
always have a table that correlates z scores and areas
under a normal curve (show example on overhead
projector)
 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.
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.




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.
Sampling error for proportions
 As N becomes large (100 or more),
the proportion in the sample is an
unbiased estimate of the proportion
in the sampling distribution (and the
population)
 The standard error of proportions =
the square root of (.25/N)
Constructing Confidence Intervals
For Proportions
 Procedures:
 Set alpha.
 Find the associated Z score.
 For an alpha of .05, we put .025 in each
tail of the normal distribution, and using
our table of normal curve areas, Z =
1.96.
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.