STA220H5 Chapter Notes - Chapter 5.3: Probability Density Function, Interquartile Range, Standard Deviation
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Probability distribution for a normal random variable x
Probability density function: f(x) = 1/a2e-(1/2)[x-u)/a]
Where
U=mean of the normal random variable x
A=standard deviation
Pie=.…
E=.…
P(x<a) is obtained from a table of normal probabilities
The standard normal distribution is a normal distribution with u=0 and a=1. A
random variable with a standard normal distribution, denoted by the symbol z, is
called a standard normal random variable.
Converting a normal distribution to a standard normal distribution
If x is a normal random variable with mean u and standard deviation a, then the
random variable z defined by the formula
Z=x-u/a
Has a standard normal distribution. The value z describes the number of standard
deviations between x and u.
Steps for finding a probability corresponding to a normal random variable
1. Sketch the normal distribution and indicate the mean of the random variable
x. then shade the area corresponding to the probability you want to find.
2. Convert the boundaries of the shaded area from x values to standard normal
random variable z values by using the formula
Z=x-u/a
Show the z values under the corresponding x values on your sketch
3. Use technology or table II in appendix b to find the areas corresponding to
the z values. If necessary, use the symmetry of the normal distribution to find
areas corresponding to negative z values and the fact that the total area on
each side of the mean equals .5 to convert the areas from table II to the
probabilities of the event you have shaded.
Determining whether the data are from an approximately normal distribution
1. Construct either a histogram or stem and leaf display for the data, and
note the shape of the graph. If the data are approximately normal, the
shape of the histogram or stem and leaf display will be similar to the
normal curve shown in figure 5.6 (i.e., the display will be mound shaped
and symmetric about the mean).
2. Compute the intervals x+s, x+2s, and x+3s, and determine the percentage
of measurements falling into each. If the data are approximately normal,
the percentages will be approximately equal to 68%, 95%, and 100%,
respectively.
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Document Summary
Probability distribution for a normal random variable x. P(x
