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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

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

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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.
Transcribed Image Text: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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