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Estimating the Population Proportion p

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 $\hat{p}=6/40=.15$ of the graduating class plan to go to graduate school. Now $\hat{p}$ 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 $\hat{p}$ equals $\sqrt{p(1-p)}/\sqrt{n}$ . It follows that the expected size of the miss is $\sqrt{p(1-p)}/\sqrt{n}$ . 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 $\hat{p}$ 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 .

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} The population proportion $p$\space is esti... ...unt called the {\em standard error} (SE) of $\hat{p}$\space . \vspace*{1ex} } }$

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} The SE of $\hat{p}$\space is calculated as $\sqrt{\hat{p}(1-\hat{p})}/\sqrt{n}$\vspace*{1ex} } }$

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 $\hat{p}$ , 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 $\hat{p}$ by the squareroot of that factor

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} As sample size increases, the SE of $\hat{p}$\space decreases like the squareroot of sample size. \vspace*{1ex} } }$

Next: Exercises Up: Sampling Distribution of the Previous: The Sampling Distribution of

2003-09-08