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Imagine a class with 100 students and at the end of the semester they
all took the final exam. After the teacher graded the exam he noticed
that a lot of students scored between 70 and 75 percent. A slightly
smaller group of students must have studied a little harder as they
got between 75 and 90 percent, and a similar sized group probably
spent more time at the local pub because they scored between 55 and 70
percent. Even rarer were the groups that aced or totally bombed the test.
If the teacher wanted to visualize how the grades on his final exam
were distributed, he could draw out a histogram (see Figure
3.2.1)that would show the
number of exams with scores that fell into different ranges.
Figure:
A histogram of exam scores

If he then scaled each column in the histogram by dividing by the
total number of exams (in this case, 100), the histogram would also
give the teacher a rough estimate of the probability of picking an
exam at random with a score between 70 and 75.
If the
teacher wanted to know what the probability the exam score would be
between 70 and 90, he could simply add the columns that represented
that range together to give himself a general idea.
If the teacher fit a curve to the scaled version of the histogram, so
that the total area under the curve was equal to 1, he
could use integration to estimate the probability of an exam having a
score between any two points. The smaller the area between two
points, the lower the probability that an exam will have that score.
The larger the area between two points, the greater the probability of
randomly selecting an exam in that range.
These probabilities are confirmed by the original histogram.
At this point, you may be wondering why the teacher would want to use
a fitted curve instead of his original histogram to determine probabilities.
The reason for this is that it is easier to compare curves, and thus,
use them to answer questions. If the teacher fit curves to several
year's worth of histograms, he could use them to determine if having
the study session two days before the exam helped or not.
Next: Comparing Two Samples: Classifying
Up: Formal Questions of a
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Frank Starmer
20040519