Estimating the Population Proportion

The TV World computations in the previous section assume that we know the warranty rate
is *p*=.20. In data analysis, population parameters like *p* are typically unknown
and estimated from the data. Consider estimating the proportion *p* of the current
WMU graduating class who plan to go to graduate school. Suppose we take a sample of
40 graduating students, and suppose that 6 out of the 40 are planning to go to graduate
school. Then our estimate is
of the graduating class plan to go to
graduate school. Now
is based on a sample, and unless we got
really lucky, chances are the .15 estimate *missed*. By how much?
On the average, a random variable misses the mean by one SD. From
the previous section, the SD of
equals
.
It follows that
the *expected size of the miss* is
.
This last term
is called the *standard error of estimation* of the sample proportion, or simply
standard error (SE) of the proportion
.

However, since we do not know *p*, we cannot calculate this SE. In a situation like this,
statisticians replace *p* with
when calculating the SE. The resulting quantity
is called the *estimated standard error of the sample proportion*
.
In practice, however, the word ``estimated'' is dropped and the estimated SE is simply called
the SE .

Exercise 4.

a. If 6 out of 40 students plan to go to graduate school, the proportion of all students who plan to go to graduate school is estimated as ________. The standard error of this estimate is ________.

b. If 54 out of 360 students plan to go to graduate school, the proportion of all students who plan to go to graduate school is estimated as ________. The standard error of this estimate is ________.

Exercise 4 shows the effect of of increasing the sample size on the SE of the sample proportion. Multiplying the sample size by a factor of 9 (from 40 to 360) makes the SE decrease by a factor of 3. In the formula for the SE of , the sample size appears (i) in the denominator, and (ii) inside a squareroot. Therefore, multiplying the sample size by a certain factor divides the SE of by the squareroot of that factor