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Probability = area under the curve - area under the curve formula


Probability = area under the curve-area under the curve formula

MAT 141 - Statistics Page 1
Section 7.2 (Sullivan 5e)
Learning Outcomes
After we cover Section 7.2, you should be able to:
1. Describe what is meant by a z-score (without describing any arithmetic operations or
writing a formula) and calculate the z-score for a given value of a random variable X.
2. Describe what is meant by a standard normal distribution.
3. Find areas (i.e., probabilities) under a (standard) normal distribution curve using Table V
and/or calculator functions.
4. Given a normally distributed random variable X, use Table V and/or calculator functions
to find:
a. The value(s) of X that bound a given percentage of the observations (e.g., the
middle 30%, the upper 10%).
b. A given percentile of the random variable X.
c. z , i.e., the z-score that is to the LEFT of a region whose area is .
5. Draw a rough sketch of the (standard) normal distribution curve and shade the area
associated with the probability being calculated. Use the Empirical Rule to draw a
reasonably accurate graph (e.g., the curve will approach the horizontal axis at about 3
standard deviations on either side of the mean).
6. Apply these techniques to real world applications involving (approximately) normally
distributed random variables.
Review: Interpreting the Area Under a Normal Probability Density Curve
Probabilities (and proportions) are equivalent to areas under the probability density curve.
PrTohibs ias btruieliftoyr an=y parorbeabailituy ndednseityrctuhrvee, ncotujursvt neormal distributions!
P(aXb)
= Proportion of observations between a and b
= Area under the curve between X=a and X=b
X
MAT 141 (Sullivan 3e) - 7.1-7.3 ? 2003 The McGraw-Hill Companies, Inc. a b GHK 03/2012
Slide 21
geoffrey.krader@morton.edu kradermath.jimdo.com 03/2018
MAT 141 - Statistics Page 2
Section 7.2 (Sullivan 5e)
At the end of Section 7.1 we started to look at the following example:
EXAMPLE: Giraffe Weights
Assume that giraffe weights are (approximately) normally distributed with a mean of 2200 lbs. and a
standard deviation of 200 lbs.
We learned that the shaded area
represents:
The probability that a randomly
selected giraffe weighs less than 2100
lbs.
The proportion of giraffes who weigh
less than 2100 lbs.
In Section 7.2 we will learn how to
calculate the shaded area.
FindingCthae Alcreua Ulandteirna NgortmhaleProabarbeiliaty DuennsdityeCurrvtehe curve
of a normal distribution
Curve #1 Curve #2 Curve #3
Curve #1 Curve #1
? 2002 PZea=rson_Ed_u_ca_t_ion, Inc. _____ _____
The figuMrAeT 1a41b(Soulvlivean4seh) -o7.2ws probability density curves for three random variablesGHoK 0n3/2t0h16e same horizontal
axis. Slide 5
Suppose we want to find the following probabilities:
For Curve #1: P(X 3)
For Curve #2: P(X 2.5)
For Curve #3: P(X 7)
In order to find these probabilities, we need to calculate the areas under each curve to the left of the
dashed line. We can calculate probabilities using a table or using calculator functions. Calculator
functions are generally faster, more accurate and easier to use, but using the tables (at least at the
beginning) may force you to learn more about the characteristics of the normal distribution!
geoffrey.krader@morton.edu kradermath.jimdo.com 03/2018
MAT 141 - Statistics Page 3
Section 7.2 (Sullivan 5e)
Finding the Area Under a Normal Curve Using a Table
To find the area (probability) under a normal probability density curve, we will use a table where:
You look up a value on the horizontal axis, and
The table tells you the area under the curve to the left of that value.
However, each normal probability density curve is defined by the mean and standard deviation
, so we would need a separate table for each curve. To see this more clearly, look at how
P(X 2.5) varies for each of the three curves on page 2. (Remember, each normal curve continues
on the right to and on the left to ):
Curve #1 P(X 2.5) is close to 1. Almost 100% of the area under this curve
is to the left of 2.5.
Curve #2 P(X 2.5) is less than 0.5. Less than 50% of the area under this curve
is to the left of 2.5.
Curve #3 P(X 2.5) is close to 0. Practically none of the area under this
curve is to the left of 2.5.
However, it would be impossible to construct a table for each normal curve because there are
infinitely many normal curves!
We can avoid creating infinitely many tables by thinking about z-scores on the horizontal axis
instead of the values of the random variable X. Recall (from Section 3.4) that the z-score describes
the position of a data point X as the number of standard deviations from the mean:
Z X

EXAMPLE: Calculate z-scores
In Curve #3 (where 9 and 2 ), find the z-score for X 7 .
By calculating the z-scores for values of X, we describe the values of the random variable relative to
the mean and the standard deviation. For example, in Curve #3:
7 is ________ standard deviation(s) to the ______________ of the mean.
Therefore, if we had a table that showed areas under the curve relative to z-scores, we could look up
the area to the left of Z= ______ to find the area to the left of X=7.
geoffrey.krader@morton.edu kradermath.jimdo.com 03/2018

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