Transcript Chapter 9: Estimation Using a Single Sample

Chapter 9: Estimation Using a
Single Sample
Section 9.1: Point Estimation
• Point Estimation – of a population characteristic
is a single number that is based on sample data
and that represents a plausible value of the
characteristic.
• A point estimate is obtained by first selecting an
appropriate statistic. The estimate is then the
value of the statistic for the given sample.
• For example: The sample mean (x bar) provides
a point estimate of a population mean μ
Example
• An article reported that 537 of the 1013
people surveyed believed that affirmative
action programs should be continued.
Let’s use this information to estimate π,
where π is the true proportion of all U.S.
adults who favor continuing affirmative
action programs.
An obvious choice for obtaining a point estimate is to use the sample
proportion of successes.
number of successes in the sample
537
p

 .530
n
1013
That is based on this random sample, we estimate that 53% of the adults
in the United States believe that affirmative action programs should be
contained.
Example
• The article “Online Extracurricular Activity”
reported the results of a study of college
students conducted by a polling
organization called the Student Monitor.
One aspect of computer use examined in
this study was the number of hours per
week spent on the Internet. Suppose that
they following observations represent the
number of Internet hours per week
reported by 20 college students:
4.00 5.00 5.00 5.25 5.50 6.25 6.25
6.50 6.50 7.00 7.25 7.75 8.00 8.00
8.25 8.50 8.50 9.50 10.50
Suppose that a point of estimation of μ, the
true mean is desired. An obvious choice to
use is sample mean, but it is not the only
choice. You may also consider the
trimmed mean or the median.
sample mean  x 
 x  141.50  7.075
n
20
7.0  7.25
sample median 
 7.125
2
10% trimmed mean  (average of middle 16 observatio ns) 
112.5
 7.031
16
The estimates of the mean Internet time per week for college students differ
somewhat from one another. The choice from among them should depend
on which statistic tends, on average, to produce an estimate closest to the
true value of μ. The following will discuss criteria for choosing among
competing statistics.
Choosing a Statistic for Computing
an Estimate
• The statistic used should be one that
tends to yield an accurate estimate – that
is an estimate close to the value of the
population characteristic. Information is
provided in the sampling distribution.
• The distribution in Figure (a) is that of a statistic unlikely
to yield an estimate close to the true value. Since the
distribution is centered to the right of the true value, an
estimate will be larger than the true value.
• For this distribution a trimmed mean would be the best
estimate.
• The distribution for figure (b) is centered at the true
value. Thus, although one estimate may be smaller than
the true value and another may be larger, in the long run
there will be no over or under estimates.
• For this distribution, the median would be a good
estimate.
• The distribution in figure (c) is exactly the true
value of the population characteristic (implying
no error in estimation).
• For this distribution the mean is the true value.
• A statistic whose mean value is equal to
the value of the population characteristic
being estimated is said to be an unbiased
statistic.
• A statistic is not unbiased is said to be
biased.
• Given the choice between several
unbiased statistics that could be used for
estimating a population characteristic, the
best statistic to use is the one with the
smallest standard deviation.