The rules of probability apply to the normal curve. For example, if we want to
create two boundaries that collectively contain five percent of the scores in
two ends of the normal curve, we could identify those boundaries using
probability. The normal curve is a graph of all of the scores in a distribution.
The most common highest probability score is the mean. Notice the mean on the
x-axis. The curve directly above it is the highest part of the curve. The curve
represents relative frequency, so the highest part of the curve is also the
part with the highest frequency. Scores closer to the mean have higher
probabilities. Scores further from the mean have lower probabilities. Scores on
the x-axis that are far from the mean have low frequency. There are very few of
those scores, so how different does the test score need to be from the
population mean for us to consider it unlikely to represent the population? We
set an arbitrary cutoff value of 5% i.e. if the test mean is so extreme that it
would occur randomly less than five times out of 100, then it is too unlikely
to represent the population mean.
The value corresponding to a probability of five percent or P equals 0.05.
is Z of a positive or negative 1.96 standard deviations. The probability of
randomly choosing a score above or below positive or negative 1.96 is p equals
.05. I will explain this in much more detail in a future chapter on
z-scores. If you want to exclude p equals .05 of your scores, you can set
a critical value to mark a region of rejection. The critical value is the five
percent cutoff value that sets the inner fence of the region of rejection. The
criterion we use is five percent, p equals .05, although we could use
other values if we wanted to. The critical value is also called the alpha
level: alpha, alpha value, alpha number, level of significance, significance
criteria, significance value, cutoff score, or that number that we look up in the
table. The region of rejection aka the critical region is the area under the
normal curve containing scores that are so unlikely that we reject the idea that
they represent our population. These are the small ends of the tails. In this case,
they're colored in purple. Means in the region of rejection are so
unrepresentative of the original population that it is a much better bet
that they represent some other population. If the test mean does not lie
beyond the critical value, and the sample is not in the region of rejection.
Therefore the sample test mean is likely to represent the original population.