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§ Continuous Probability §
Introduction- Lesson 1
Continuous probability distributions share the same characteristics with continuous data. Discrete data are data which have "gaps" between them due to physical or logical reasons. Continuous data are data which have any possible values of a range. Continuous probability distributions include continuous data and probabilities associated with ranges of those data. One can study several types of continuous probability distributions including the normal (bell shaped), uniform, negative exponential among others. For the purposes of this course, you will study the normal probability distribution.
§ Standard Normal Distribution §
Normal Distribution
The normal distribution has bell
shape. The mean of a normal distribution is m and
a standard deviation of s. The
area under the curve is equal to 1 (100%). The distribution is symmetrical with
50% of the data value to the left of m and
50% to the right. The probability of observing a single data point in a continuous
distributions is zero. A range of data values is necessary to calculate probabilities.
It is not easy to calculated these probabilities with data on x. One must integrate
(using calculus) the specific equation for the curve under consideration over
a range of x values. We use the Z distribution, the Standard Normal Distribution,
to obtain the probabilities without the use of calculus. In fact, the Z
table gives these results. 
(click me)
If in a normal distribution m equals 325, what is the probability that the
data value x is less than 325? 1.00
0.50
can
not determine (click one)
Standard Normal Distribution
The standard normal distribution
has a bell shape. The mean of a normal distribution is 0 and a standard deviation
of 1. The area under the curve is equal to 1 (100%). The distribution is symmetrical
with 50% of the data values to the left of m and
50% to the right. The probability of observing a single data point in a continuous
distributions is zero. A range of data values is necessary to calculate probabilities.
One must convert (standardize or normalize) any x data value in a problem to
a Z score. The process is simple: Z = (x - m)/s; that is, subtract the
mean from the x data value and divide the result by the standard deviation.
The numerator of this fraction is an "individual deviation"
and the denominator is standard or "average deviation." The
Z score counts the number of standard deviations that x is away from
m. All
Z scores on the right side of the distribution are positive, the middle one
zero and all on the left side negative. Once the Z score is obtained, then the
Z table is used to find probabilities.
Any probabilities found for Z scores, correspond to the probabilities for the
individual x data values used to calculate the Z scores.
(click me)
If in a normal distribution m equals 325, what is the Z score associated
with an x value of 325? zero
one
can
not determine (click
one)
Tables for Z Scores
Z tables may be constructed in several
different ways. But in general, all Z tables have probabilities in the body
and Z scores around the edges. Usually the first part of the Z score is found
on the left edge (e.g. 2.3) and the second digit to the right of the decimal
point across the top (e.g. .07). Thus, this example would be for the Z score
2.37. The Z table
found at this web site shows only positive Z scores; but, since the standard
normal distribution is symmetrical, positive Z scores give the same probabilities
as for negative Z scores. This table gives the probability of being at a specific
Z score position to the middle (Z = 0). Other printed tables provide both positive
and negative Z scores. In general, such tables accumulate probabilities from
left to right in the distribution (always gives a left tail as the probability)
up to the position of the Z score. In fact, the default for Excel is this type
of table.
(click me)
A Z score of - 1.33 gives a probability the same as the Z score 1.33. True
False
(click one)