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 .
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