Download SOC 2105 – ELEMENTS OF SURVEY SAMPLING AND SOCIAL

SOC 2105 – ELEMENTS OF SURVEY SAMPLING AND SOCIAL STATISTICS
LECTURE NOTES
LECTURE SEVEN – November 12, 2015
Estimation is a procedure by which a numerical value or values are assigned to a population
parameter based on the information collected from a sample.
In inferential statistics, µ is called the true population mean and p is called the true population
proportion. There are many other population parameters, such as the median, mode, variance,
and standard deviation.
The following are a few examples of estimation: an auto company may want to estimate the
mean fuel consumption for a particular model of a car; a manager may want to estimate the
average time taken by new employees to learn a job; the U.S. Census Bureau may want to find
the mean housing expenditure per month incurred by households; and the AWAH (Association
of Wives of Alcoholic Husbands) may want to find the proportion (or percentage) of all
husbands who are alcoholic.
The examples about estimating the mean fuel consumption, estimating the average time taken to
learn a job by new employees, and estimating the mean housing expenditure per month incurred
by households are illustrations of estimating the true population mean, µ . The example about
estimating the proportion (or percentage) of all husbands who are alcoholic is an illustration of
estimating the true population proportion, p.
The value(s) assigned to a population parameter based on the value of a sample statistic is called
an estimate of the population parameter. For example, suppose the manager takes a sample of 40
new employees and finds that the mean time, taken to learn this job for these employees is 5.5
hours. If he or she assigns this value to the population mean, then 5.5 hours is called an estimate
of 𝑋. The sample statistic used to estimate a population parameter is called an estimator. Thus,
the sample mean, is an estimator of the population mean, 𝑋; and the sample proportion, pˆ, is an
estimator of the population proportion, p.
Definition
Estimate and Estimator The value(s) assigned to a population parameter based on the value of
a sample statistic is called an estimate. The sample statistic used to estimate a population
parameter is called an estimator.
The estimation procedure involves the following steps.
1. Select a sample.
2. Collect the required information from the members of the sample.
3. Calculate the value of the sample statistic.
4. Assign value(s) to the corresponding population parameter.
Remember, the procedures to be learned in this chapter assume that the sample taken is a
simple random sample. If the sample is not a simple random sample, then the procedures to be
used to estimate a population mean or proportion become more complex.
Point and Interval Estimates
A Point Estimate
If we select a sample and compute the value of the sample statistic for this sample, then this
value gives the point estimate of the corresponding population parameter.
Definition
Estimate and Estimator The value(s) assigned to a population parameter based on the value of
a sample statistic is called an estimate. The sample statistic used to estimate a population
parameter
is called an estimator.
Definition
Point Estimate The value of a sample statistic that is used to estimate a population parameter is
called a point estimate.
Thus, the value computed for the sample mean, from a sample is a point estimate of the
corresponding population mean. For the example mentioned earlier, suppose the Census Bureau
takes a sample of 10,000 households and determines that the mean housing expenditure per
month, for this sample is $1970. Then, using 𝑥as a point estimate of µ, the Bureau can state that
the mean housing expenditure per month, µ, for all households is about $1970. Thus,
Point estimate of a population parameter = Value of the corresponding sample statistic.
Each sample selected from a population is expected to yield a different value of the sample
statistic. Thus, the value assigned to a population mean, µ, based on a point estimate depends on
which of the samples is drawn. Consequently, the point estimate assigns a value to µ that almost
always differs from the true value of the population mean.
An Interval Estimate
In the case of interval estimation, instead of assigning a single value to a population parameter,
an interval is constructed around the point estimate, and then a probabilistic statement that this
interval contains the corresponding population parameter is made.
For the example about the mean housing expenditure, instead of saying that the mean housing
expenditure per month for all households is $1970, we may obtain an interval by subtracting a
number from $1970 and adding the same number to $1970. Then we state that this interval
contains the population mean, µ. For purposes of illustration, suppose we subtract $340 from
$1970 and add $340 to $1970. Consequently, we obtain the interval ($1970 – $340) to ($1970 +
$340), or $1630 to $2310. Then we state that the interval $1630 to $2310 is likely to contain the
population mean, µ and that the mean housing expenditure per month for all households in the
United States is between $1630 and $2310. This procedure is called interval estimation. The
value $1630 is called the lower limit of the interval, and $2310 is called the upper limit of the
interval. The number we add to and subtract from the point estimate is called the margin of
error. Figure 8.1 illustrates the concept of interval estimation.
Estimation of a population mean when the Standard Deviation is known
Determining the Sample Size for the Estimation of Mean
An alumni association wants to estimate the mean debt of this year’s college graduates. It is
known that the population standard deviation of the debts of this year’s college graduates is
$11,800. How large a sample should be selected so that the estimate with a 99% confidence level
is within $800 of the population mean?
Solution The alumni association wants the 99% confidence interval for the mean debt of this
year’s college graduates to be
x ± 800