See the photo for the examples.A. Determine the area BELOW the following:1. z = 3.12. z = 1.673. z = 0.76B. Determine the area ABOVE the following:1. z = -2.52. z = 1.963. z = -0.15C. Determine the area of the region indicated:1. -1 < z < 12. -1.5 < z < 2.53. -2.33 < z < 1.64 The Standard Normal CurveSince the appearance of the normal curve depends on the distribution's meanand standard deviation, there must be infinitely many different normal curves. In orderto make use of its properties, statisticians came up with a way of transforming everynormal curve to a standard normal distribution/curve.A standard normal distribution is a normal distribution with mean of 0 andstandard deviation of 1. Basically, any normal distribution could be transformed intothis type.The formula for the standard normal distribution isy = anThe figure below illustrates a standard normal distribution. The values at thehorizontal axis are the values of the random variable z, the transformed values of therandom variable X. The values of z are computed using the formula:value - meanor z= X-4standard deviation0.13%13%2.19%2.13%13.50% 34.1% 34.1313.59%30-20-lolo+20+30-2-1+1vahuesEmpirical RuleThe empirical rule is based on the equation introduced by de Moivre for anynormally distributed random variable X.1. Approximately 68.26% of all values of X lie within one standard deviationfrom the mean u, i.e. approximately this percent of all values lie in theinterval (u - o, u + o) with about 34.13% on each side of the mean.2. Approximately 95.44% of all values lie within two standard deviationsfrom the mean, or within the interval (u - 20 , u + 20).3.About 99.74% of all observations lie within three standard deviations fromthe mean, or within the interval (u - 30 , u+ 30).Finding the Areas under the Standard Normal CurveThe most important part of the normal curve is the area bounded by somevalues. A table (z-Table: Areas under the Normal Curve) was included at the lastpages to make the calculation easier. A four-decimal-place number in the table givesthe area under the standard normal curve from -3.9 to +3.9 of z-values/scores.Examples1. Determine the area under the standard normal curve between 0 and 1.97.Area1.97-1 0 +1 +2-3-21. Locate the 1.9 on the leftmost column of the Z-table, and .07 on the topmostrow. Their intersection is 0.9756.2. Locate 0.0 on the leftmost column, and .00 on the topmost row. Theirintersection is .5000.%3D3. To get the area between them, subtract as follows: 0.9756 -0.5000 = 0.4756.02.07 .08.09.00.01.03.04 .05.060.0 L5000 .5040 .5080 5120 5160 5199 5239 .5279 .5319 .53590.1.5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .57530.21.8 .9641 .9649 .9656 .9664 9671 .9678 .9686 .9693 .9699 .97061.9 | .9713 .9719 .9726 .9732 .9738 .9744 9750 9156 9761 .9767The area under the standard normal curve between 0 and 1.97 is 0.4756.

Answered: See the photo for the examples. A.…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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See the photo for the examples. A. Determine the area BELOW the following: 1. z = 3.1 2. z = 1.67 3. z = 0.76 B. Determine the area ABOVE the following: 1. z = -2.5 2. z = 1.96 3. z = -0.15 C. Determine the area of the region indicated: 1. -1 < z < 1 2. -1.5 < z < 2.5 3. -2.33 < z < 1.64

The Standard Normal Curve
Since the appearance of the normal curve depends on the distribution's mean
and standard deviation, there must be infinitely many different normal curves. In order
to make use of its properties, statisticians came up with a way of transforming every
normal curve to a standard normal distribution/curve.
A standard normal distribution is a normal distribution with mean of 0 and
standard deviation of 1. Basically, any normal distribution could be transformed into
this type.
The formula for the standard normal distribution is
y = an
The figure below illustrates a standard normal distribution. The values at the
horizontal axis are the values of the random variable z, the transformed values of the
random variable X. The values of z are computed using the formula:
value - mean
or z= X-4
standard deviation
0.13%
13%
2.19%
2.13%
13.50% 34.1% 34.13
13.59%
30
-20
-lo
lo
+20
+30
-2
-1
+1
vahues
Empirical Rule
The empirical rule is based on the equation introduced by de Moivre for any
normally distributed random variable X.
1. Approximately 68.26% of all values of X lie within one standard deviation
from the mean u, i.e. approximately this percent of all values lie in the
interval (u - o, u + o) with about 34.13% on each side of the mean.
2. Approximately 95.44% of all values lie within two standard deviations
from the mean, or within the interval (u - 20 , u + 20).
3.
About 99.74% of all observations lie within three standard deviations from
the mean, or within the interval (u - 30 , u+ 30).
Finding the Areas under the Standard Normal Curve
The most important part of the normal curve is the area bounded by some
values. A table (z-Table: Areas under the Normal Curve) was included at the last
pages to make the calculation easier. A four-decimal-place number in the table gives
the area under the standard normal curve from -3.9 to +3.9 of z-values/scores.
Examples
1. Determine the area under the standard normal curve between 0 and 1.97.
Area
1.97
-1 0 +1 +2
-3
-2
1. Locate the 1.9 on the leftmost column of the Z-table, and .07 on the topmost
row. Their intersection is 0.9756.
2. Locate 0.0 on the leftmost column, and .00 on the topmost row. Their
intersection is .5000.
%3D
3. To get the area between them, subtract as follows: 0.9756 -0.5000 = 0.4756
.02
.07 .08
.09
.00
.01
.03
.04 .05
.06
0.0 L5000 .5040 .5080 5120 5160 5199 5239 .5279 .5319 .5359
0.1
.5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
0.2
1.8 .9641 .9649 .9656 .9664 9671 .9678 .9686 .9693 .9699 .9706
1.9 | .9713 .9719 .9726 .9732 .9738 .9744 9750 9156 9761 .9767
The area under the standard normal curve between 0 and 1.97 is 0.4756.

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