Session 10 Notes, Section 6-4
The Probability of a Normal Distribution
In Session 10 we learn how to find the probability of a normal distribution
and to determine a random variable value that is associated with a given
probability.
At the end of this session you will be able to:
If you have learned all of these objectives, then close this window to
return to where you were.
- The z-value is the number of standard deviations that a
particular X value is away from the mean. The formula for
finding the z value is:
- To find the area between 0 and any z value: Look up the z
value in the table to get the area. notice in your
textbook, that Table E, found on page 730 or on your Formula
Card, gives the area between z = 0 and z = a positive
value for z! In using the table, always look for a
diagram, such as you see at the bottom of page 730 or the
bottom of your Formula Card, that shows the area found in the
table.
- Area in Any Tail
- Look up the z value to get the area.
- Subtract the area from 0.5000.
- Area Between Two z Values
- Look up both z values to get the areas.
- Subtract the smaller area from the larger area.
- Area Between z Values—Opposite Sides of the mean
- Look up both z values to get the areas.
- Add the areas.
- Area To the Left of Any z Value, where z is great than the
mean.
- Look up the z value to get the area.
- Add 0.5000 to the area.
- Area To the Right of Any z Value
- Look up the z value in the table to get the area, where z
is less than the mean..
- Add 0.5000 to the area.
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- Area Under the Curve
- The area under the curve is important
because the area is the same as the probability,
for a particular value of the random variable.
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We work backwards from what we did to
find probabilities.
- Using Table E and the given probability, we find the closest value
"inside" Table E.
- Find the Z-value at the left of the Table E corresponding to the
probability in Table E.
- Solve for the X random variable value using algebra and the
formula z = (X - μ) /
σ.
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- The normal distribution can be used to describe a variety
of variables, such as heights, weights, and temperatures.
- The normal distribution is bell-shaped, unimodal,
symmetric, and continuous; its mean, median, and mode are
equal.
- Mathematicians use the standard normal distribution which
has a mean of 0 and a standard deviation of 1.
- The normal distribution can be used to describe a sampling
distribution of sample means.
- These samples must be of the same size and randomly
selected with replacement from the population.
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When you finish these notes, then close this window to return to where
you were.