http://www.ck12.org Chapter 7. Sampling Distributions and Estimations
administration. The study population will be the 18,000 students that attend the school. The elements will be the
individual students. A random sample of 100 students will be selected for the purpose of estimating the entire student
body. Attitudes toward the dress code will be the variable under consideration. For simplicity sake, assume that the
attitude variable has two attributes: approve and disapprove. As you know from the last chapter, in a scenario such
as this when a variable has two attributes it is calledbinomial.
The following figure shows the range of possible sample study results. The horizontal axis presents all possible
values of the parameter in question. It represents the range from 0 percent to 100 percent of students approving of
the dress code. The number 50 on the axis represents the midpoint, 50 percent, of the students approving the dress
code and 50 percent disapproving. Since the sample size is 100, half of the students are approving and the other half
are disapproving.
To randomly select the sample of 100, every student is presented with a number (from 1 to 18,000) and the sample
is randomly selected from a drum containing all of the numbers.
Each member of the sample is then asked whether they approve or disapprove of the dress code. If this procedure
gives 48 students who approve of the code and 52 who disapprove, the result is recorded on the horizontal axis by
placing a dot at 48%. This percentage describes the variable and is called a statistic.
Let’s assume that the process was repeated again and this resulted in 52 students approving the dress code. A third
sample resulted in 51 students approving the dress code.
In the figure above, the three different sample statistics representing the percentages of students who approved the
dress code are shown. The three random samples chosen from the population, give estimates of the parameter that
exists in the total population. In particular, each of the random samples gives an estimate of the percentage of
students in the total student body of 18,000 that approve of the dress code. Assume for simplicity that the true mean
for the entire population is 50%. Then this estimate is close to the true mean. To precisely compute the true mean,
it would be necessary to continue choosing samples of 100 students and to record all of the results in a summary
graph.
By increasing the number of samples of 100, the range of estimates provided by the sampling process has increased.
It looks as if the problem in attempting to guess the parameter in the population has also become more complicated.
However, probability theory provides an explanation of these results.