lecture 7 presentation

8/12/2019 Lecture 7 Presentation

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Class 7

Rafael Mendoza-Arriaga

Rafael Mendoza McCombs

Elementary Business Statistics Class 7


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Weighted Sums of Random Variables

Let X1, X2, . . . , X n be any (independent or dependent) randomvariables, and let a1, a2, . . . , an be any n constants. Suppose Y is the

weighted sum of the Xs;

Y =a1X1+ a2X2+ . . . + anXn




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Let X1, X2, . . . , X n be any (independent or dependent) randomvariables, and let a1, a2, . . . , an be any n constants. Suppose Y is theweighted sum of the Xs;

Y =a1X1+ a2X2+ . . . + anXn

Expected value of a weighted sum of random variables

E(Y) =a1E(X1) + a2E(X2) + . . . + anE(Xn)




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Y =a1X1+ a2X2+ . . . + anXn


E(Y) =a1E(X1) + a2E(X2) + . . . + anE(Xn)

Variance of a weighted sum of independent random variables

V ar(Y) =a21

V ar(X1) + a2

2V ar(X2) + . . . + a

2

nV ar(Xn)




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Y =a1X1+ a2X2+ . . . + anXn


E(Y) =a1E(X1) + a2E(X2) + . . . + anE(Xn)

Variance of a weighted sum of independent random variables

V ar(Y) =a21V ar(X1) + a22V ar(X2) + . . . + a2nV ar(Xn)

Variance of a weighted sum of 2 dependent random variables

V ar(Y) =a21

V ar(X1) + a2

2V ar(X2) + 2a1a2Cov(X1, X2)




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Example I

There are two investment options:

GM StocksGold

The state of the economy is uncertain: depression, recession, normaland boom.

Depending on the state of the economy, the returns for investmentschange.

Download GM vs Gold.xlsx

Calculate the mean and standard deviation of returns from GMStocks and gold.

What is the covariance and correlation between GM Stocks andGold?

Simulate returns from GM stocks and Gold. Calculate the averageand the standard deviation of returns in your sample. Also, calculatethe covariance and correlation in your sample.




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Example II

Assume the investor has $10,000. Calculate the mean and standarddeviation of the returns from a portfolio that invests 60% of the

budget into GM Stocks and the rest into gold.Can you calculate the mean and standard deviation of the returnsfrom a portfolio that invests 70% of the budget into GM stocks?




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Apple Stock Price (12/9/0812/9/09)

100.06 88.36 83.11 121.76 140.95 147.52 169.45 190.47 199.92

98.21 89.64 88.63 121.51 143.74 151.75 170.05 190.81 205.8895 90.73 92.68 125.4 144.67 152.91 168.21 190.02 204.4498.27 94.2 96.35 123.9 143.85 151.51 165.3 191.29 204.1994.75 93 95.93 124.73 142.72 156.74 165.18 190.56 200.5995.43 90.13 95.42 123.9 140.25 157.82 166.55 188.05 199.9189.16 91.51 99.66 125.14 139.95 159.99 170.31 189.86 196.9789.43 92.98 101.52 125.83 136.97 160.1 172.93 198.76 196.23

90 93.55 101.62 127.24 136.09 160 171.14 204.92 196.4885.74 96.46 101.59 132.07 136.35 160.03 172.56 205.2 193.3286.38 99.72 107.66 132.71 135.58 162.79 172.16 203.94 188.9585.04 102.51 106.5 132.5 135.88 163.39 173.72 202.48 189.87

85.81 97.83 106.49 129.06 139.48 166.43 175.16 197.3786.61 96.82 109.87 129.19 137.37 165.55 181.87 192.486.29 99.27 106.85 129.57 134.01 165.11 184.55 196.3585.35 99.16 104.49 124.42 136.22 163.91 185.02 188.590.75 94.53 105.12 119.49 139.86 165.51 184.02 189.3194.58 94.37 108.69 122.95 142.44 164.72 184.48 188.7593.02 90.64 112.71 122.42 141.97 162.83 185.5 190.8191.01 91.2 115.99 126.65 142.43 165.31 183.82 194.0392.7 86.95 118.45 127.45 142.83 168.42 182.37 194.34

90.58 90.25 115 125.87 140.02 166.78 186.15 201.4688.66 91.16 116.32 124.18 138.61 159.59 185.38 202.9887.71 89.19 119.57 122.5 135.4 164 185.35 203.2585.33 89.31 120.22 130.78 137.22 164.6 180.86 201.9983.38 87.94 118.31 133.05 136.36 166.33 184.9 204.4582.33 88.37 117.64 135.07 138.52 169.22 186.02 206.6378.2 91.17 121.45 135.81 142.34 169.06 190.01 207

82.83 88.84 123.42 139.35 142.27 169.4 190.25 205.9688.36 85.3 120.5 139.49 146.88 167.41 189.27 200.51 ($)




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Apple Stock Price (12/9/0812/9/09)

Time Plot: Apple Stock Prices

Jan Apr Jul Oct

80

100

120

140

160

180

200




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Calculating returns

AReturnis the Stock PricePercentage Change,

ri = 100 Si Si1

Si1

For example,The Apples stock price on December 8, 2009 was $188.95. On the nextday (Dec. 09) the stock price moved to $189.87, this represents a 0.49%increase in price in one day,

r= 100189.87 188.95

188.95 = 0.49%




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Apple Stock Returns (12/10/0812/9/09)

-1.85 1.45 6.64 -0.21 1.98 2.87 0.35 0.18 2.98

-3.27 1.22 4.57 3.20 0.65 0.76 -1.08 -0.41 -0.703.44 3.82 3.96 -1.20 -0.57 -0.92 -1.73 0.67 -0.12-3.58 -1.27 -0.44 0.67 -0.79 3.45 -0.07 -0.38 -1.760.72 -3.09 -0.53 -0.67 -1.73 0.69 0.83 -1.32 -0.34-6.57 1.53 4.44 1.00 -0.21 1.37 2.26 0.96 -1.470.30 1.61 1.87 0.55 -2.13 0.07 1.54 4.69 -0.380.64 0.61 0.10 1.12 -0.64 -0.06 -1.04 3.10 0.13-4.73 3.11 -0.03 3.80 0.19 0.02 0.83 0.14 -1.610.75 3.38 5.97 0.48 -0.56 1.72 -0.23 -0.61 -2.26-1.55 2.80 -1.08 -0.16 0.22 0.37 0.91 -0.72 0.490.91 -4.57 -0.01 -2.60 2.65 1.86 0.83 -2.52

0.93 -1.03 3.17 0.10 -1.51 -0.53 3.83 -2.52-0.37 2.53 -2.75 0.29 -2.45 -0.27 1.47 2.05-1.09 -0.11 -2.21 -3.97 1.65 -0.73 0.25 -4.006.33 -4.67 0.60 -3.96 2.67 0.98 -0.54 0.434.22 -0.17 3.40 2.90 1.84 -0.48 0.25 -0.30-1.65 -3.95 3.70 -0.43 -0.33 -1.15 0.55 1.09-2.16 0.62 2.91 3.46 0.32 1.52 -0.91 1.691.86 -4.66 2.12 0.63 0.28 1.88 -0.79 0.16-2.29 3.80 -2.91 -1.24 -1.97 -0.97 2.07 3.66-2.12 1.01 1.15 -1.34 -1.01 -4.31 -0.41 0.75-1.07 -2.16 2.79 -1.35 -2.32 2.76 -0.02 0.13-2.71 0.13 0.54 6.76 1.34 0.37 -2.42 -0.62-2.29 -1.53 -1.59 1.74 -0.63 1.05 2.23 1.22-1.26 0.49 -0.57 1.52 1.58 1.74 0.61 1.07-5.02 3.17 3.24 0.55 2.76 -0.09 2.14 0.185.92 -2.56 1.62 2.61 -0.05 0.20 0.13 -0.506.68 -3.98 -2.37 0.10 3.24 -1.17 -0.52 -2.650.00 -2.57 1.05 1.05 0.44 1.22 0.63 -0.29 (%)




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Apple Stock Returns (12/10/0812/9/09)

Time Plot: Apple Stock Returns

Jan Apr Jul Oct

0.06

0.04

0.02

0.00

0.02

0.04

0.06




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Probability Interpretation of a Histogram

Return Distribution

0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25

What is the probabilitythat buying one share of Apple I will makemore than 4%the next day?




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Return Distribution

0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25

What is the probabilitythat buying one share of Apple I will makemore than 4%the next day?

P[r >4%] = 1.59% + 0.8% + 1.59% = 3.98%




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Return Distribution

0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25

What is the probabilitythat buying one share of Apple I willloosemoneythe next day?




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Return Distribution

0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25

What is the probabilitythat buying one share of Apple I willloosemoneythe next day?

P[r


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Probability Density Curves

A smooth curve may be a good description of the overall pattern of thedata.

0.400.401.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.801.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25




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Notation

For density curves we use the Greek letters and for the meanand standard deviation.

For data that weve collected we use x and s for the mean andstandard deviation




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Probability Density Curve

Total Areaunder the curve equal to1




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Areas represent proportions




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Areas represent proportions

Probability Distributions are generally described by:

mean ()standard deviation ()




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Mean

The interpretation of or x

mean

center

balance point

Mean: Balance Point




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Median

The median of a density curve is the equal areas point

Median: Point in which both sides have the same Area

Notice that for Symmetric Distributions the Mean and Median coincide.




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Normal Distribution, a.k.a. Gaussian in honor of Gauss

Normal Distribution fitted to data.

0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25




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0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25

Characteristics of the Normal Distribution:




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0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25


Symmetric




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0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25


Symmetric

Single-peaked




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0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25


Symmetric

Single-peaked

Bell-shaped




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Why is it called Normal?




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Normal Distributions

ForNormal Distributionsthe exact density is given by twomeasures1 mean (Center Measure)

2 standard deviation (Spread or dispersion measure)

Rafael Mendoza McCombsElementary Business Statistics Class 7
http://demonstrations.wolfram.com/TheNormalDistribution/http://demonstrations.wolfram.com/TheNormalDistribution/


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For other distributions these two measures are not enough!



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As the curve becomes wider .



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Point of curvature change is located one standard deviation ()away from the mean ()



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Point of curvature change is located one standard deviation ()away from the mean ()

Applet: http://demonstrations.wolfram.com/TheNormalDistribution/

http://demonstrations.wolfram.com/TheNormalDistribution/http://demonstrations.wolfram.com/TheNormalDistribution/http://demonstrations.wolfram.com/TheNormalDistribution/http://demonstrations.wolfram.com/TheNormalDistribution/


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The 68-95-99.7 Rule

68-95-99.7 Rule. Here, = 0 and = 1,Standard Normal Dist.


Th 68 95 99 7 R l


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The 68-95-99.7 Rule


P[ < x < +] = 68%,(Here, P[1< x


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The 68-95-99.7 Rule


P[ < x < +] = 68%,(Here, P[1< x


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The 68-95-99.7 Rule


P[ < x < +] = 68%,(Here, P[1< x


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The 68 95 99 7 Rule Example


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The 68-95-99.7 Rule Example

Heights of young (American) men are approximately normal = 69 and= 2.5

68% are between 69-2.5 and 69+2.5 (66.5 to 71.5)




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95% are between 69-5 and 69+5 (64 to 74)




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95% are between 69-5 and 69+5 (64 to 74)

99.7% are between 69-7.5 and 69+7.5 (61.5 to 76.5)


The 68-95-99 7 Rule Example


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68% are between 69-2.5 and 69+2.5 (66.5 to 71.5)95% are between 69-5 and 69+5 (64 to 74)

99.7% are between 69-7.5 and 69+7.5 (61.5 to 76.5)

Im 67 so Im still in the centered range!!


Questions


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Questions

Approximately what percent of young men are taller than 74 inches?

What percent of young men are shorter than 66.5?


Questions


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Questions


2.5%



Questions


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Questions


2.5%


16% (so I guess that kind of makes me short, right?)


Standardizing


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Standardizing

x is an observation from a distribution with mean and standarddeviation


Standardizing


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g


z is the standardized value:

z= x


Standardizing


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g


z is the standardized value:

z= x

The standardized value is called the z-score


The Standard Normal


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The standard normal distribution N(0, 1) has = 0 and = 1


The Standard Normal


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Ifx is N(, ), then the standardized variable, z,



The Standard Normal


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Ifx is N(, ), then the standardized variable, z,

z= x

has the standard normal distribution



The Standard Normal N(, ) =N(0, 1)


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Apples Return Dist. has x= 0.28%and s= 2.214%

0.400.40

1.99

3.19

8.37

11.16

19.12

23.11

13.55

7.177.57

1.590.80

1.59

6 4 2 0 2 4 6

0.05

0.10

0.15

0.20

0.25

Normal Dist. (Red)has = 0.28%and = 2.214%, i.e, N(0.28, 2.214)





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Apples Return Dist. has x= 0.28%and s= 2.214%(Zoomed Out)

6 4 2 0 2 4 6

0.1

0.2

0.3

0.4






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6 4 2 0 2 4 6

0.1

0.2

0.3

0.4

Normal Dist. (Red)has = 0.28%and = 2.214%, i.e, N(0.28, 2.214)Standard Normal Dist. (Blue)has = 0%and = 1%, i.e, N(0, 1)





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6 4 2 0 2 4 6

0.1

0.2

0.3

0.4


Standard Normal Dist. (Blue)has = 0%and = 1%, i.e, N(0, 1)TheRed Histogram, is the histogram of the standardized data.

zi = xi



Another Interpretation of Standardizing


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It helps to clearly seehow many standard deviations a given point is away

from the mean.

Example: Mens heights N(69, 2.5)





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from the mean.


Sayxi= 6 tall man, its z-score is

zi= xi

= 72 69

2.5 = 1.2





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from the mean.



zi= xi

= 72 69

2.5 = 1.2

His height is 1.2 standard deviations above average





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from the mean.



zi= xi

=

72 69

2.5 = 1.2


55 tall mans z-score is

zi = xi

=65 69

2.5 = 1.6





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from the mean.



zi=

xi

=

72 69

2.5 = 1.2


55 tall mans z-score is

zi =

xi

=

65 69

2.5 = 1.6

His height is 1.6 standard deviations below average



Excel: The Normal Distribution


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Prob. Density Function

(function or curve f(x))

4 2 2 4

0.1

0.2

0.3

0.4

x

Cummulative Dist. Function

(area under the curve f(x))

4 2 2 4

0.1

0.2

0.3

0.4

x



Excel: The Normal Distribution


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Prob. Density Function

(function or curve f(x))

4 2 2 4

0.1

0.2

0.3

0.4

x

f(x) = e

(x)2

22

2 =NORMDIST(X,,, 0)

= 1

2

x

e

(u)2

22 du

Cummulative Dist. Function

(area under the curve f(x))

4 2 2 4

0.1

0.2

0.3

0.4

x

x

f(u)du= NORMDIST(X,,, 1)

= 1

2

x

e

(u)2

22 du



Excel Functions: Normal Distribution


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NORMDIST(x, mean, standard dev, cumulative)

X is the value for which you want the distribution.

Mean is the arithmetic mean of the distribution.

Standard dev is the standard deviation of the distribution.

Cumulative is a logical value that determines the form of the function.

If cumulative is TRUE=1, NORMDIST returns thecumulativedistribution function;if FALSE=0, it returns theprobability density function.



Excel: The Standard Normal Distribution (= 0and = 1)


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Function NORMDIST(x, mean, standard dev, cumulative) =NORMDIST(z, 0, 1, 1)

This function ofz is the area under the standard normal curve to the left ofz

Table entry is area to the left ofz!

4 2 2 4

0.1

0.2

0.3

0.4

z

P[random variable z] =Area under the curve up to z



Example: The Standard Normal Distribution


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The proportion of men with heights 6 (72) or below

4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849

P[Height 1.2] = 88.5% (Area under the curve up to z)

Usingx (i.e., non standardized values) x= 72, = 69 and = 2.5NORMDIST(x, , , cumul.) =NORMDIST(72, 69, 2.5, 1) = 88.5%

Using the z-score (z= (x )/ = 1.2), x= 1.2, = 0 and = 1NORMDIST(x, , , cumul.) =NORMDIST(1.2, 0, 1, 1) = 88.5%



Example:The Standard Normal Distribution


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The proportion of men with heights 6 (72) or above

4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849

P[Height >1.2] = 100 P[Height 1.2] = 11.5%





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4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849

P[Height >1.2] = 100 P[Height 1.2] = 11.5%

Because the Area under the Curve is equal to 1 (i.e., 100%)!!





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4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849

P[Height >1.2] = 100 P[Height 1.2] = 11.5%

Because the Area under the Curve is equal to 1 (i.e., 100%)!!

Again, note that excel will give you the Left tail, thus to obtain toobtain the right tail we need to use this property!





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The proportion of men with heights 6 exactly

4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849

P[Height= 1.2] = 0%, Why?





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The proportion of men with heights 6 exactly

4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849

P[Height= 1.2] = 0%, Why?

Because the area under a single point is 0!!!



Normal Calculations


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Expect questions like:

What proportions are above y?

What proportions are below x?



Normal Calculations


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What proportions are between x and y?



Normal Calculations


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What proportions are between x and y?

What proportions are outside of the range from x to y?



Examples

What proportion of young men are shorter than 55? (recall the


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What proportion of young men are shorter than 5 5 ? (recall thez-score?)



Examples

What proportion of young men are shorter than 55? (recall the


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What proportion of young men are shorter than 5 5 ? (recall thez-score?)

Proportion of young men are shorter than 55 = 5.48%

4 2 2 4

0.1

0.2

0.3

0.4

z 1.6

Area 0.0548

In Excel (and normalizing):

P(X 65) =P

z = 65 69

2.5

= P(z 1.6) =Normdist(1.6, 0, 1, 1)



Examples


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What proportion of young men are between 55 and 6? (recall thez-score?)



Examples


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What proportion of young men are between 55 and 6? (recall the

z-score?)

Proportion of young men are between 55 and 6 = 83.01%

4 2 2 4

0.1

0.2

0.3

0.4

Total Area 0.8301

a 1.6 b 1.2

How to obtain it?



Examples

Proportion of young men are


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Proportion of young men areshorter than 6 = 88.49%

4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849



Examples



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4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849


4 2 2 4

0.1

0.2

0.3

0.4

z 1.6

Area 0.0548



Examples



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4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849


4 2 2 4

0.1

0.2

0.3

0.4

z 1.6

Area 0.0548

=

Proportion of young men areshorter than between 55 and6 = 83.01%

4 2 2 4

0.1

0.2

0.3

0.4Total Area 0.8301

a 1.6 b 1.2



Examples



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4 2 2 4

0.1

0.2

0.3

0.4

z 1.2

Area 0.8849


4 2 2 4

0.1

0.2

0.3

0.4

z 1.6

Area 0.0548

=

Proportion of young men areshorter than between 55 and6 = 83.01%

4 2 2 4

0.1

0.2

0.3

0.4

Total Area 0.8301

a 1.6 b 1.2

z is after standardizing X,

P(65 X 72) =P(1.6 z 1.2)

= P(z 1.2) P(z 1.6)

In Excel:= Normdist(1.2, 0, 1, 1)Normdist(1.6, 0, 1, 1)



Backward Calculations: Inverse Functions


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We are interested in calculating the outcome for a given probabilityfor the Normal Distribution, using Excel





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In other words:1 Give me a Probability value (p)2 I will give you the value ofz that corresponds to that probability,

i.e., Ill give you

z

such that P(z z) =p





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In other words:1 Give me a Probability value (p)2 I will give you the value ofz that corresponds to that probability,

i.e., Ill give you

z

such that P(z z) =p

Recall: p corresponds to the left tail area



Excel Functions: Backward calculations

NORMINV(probability, mean, standard dev)


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Probabilityis a probability corresponding to the normal distribution.





Excel Functions: Backward calculations

NORMINV(probability, mean, standard dev)


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Probabilityis a probability corresponding to the normal distribution.



You can use the distribution ofx (non-standardized value) or z the standardnormal

x =N ORMIN V(p, , )z =N ORMIN V(p, 0, 1) and then to obtain x = +z (sincez = (x )/))



Backward Calculations


89/95

Find the value with a given proportion of the observations above (or below) it





90/95


1 State the problem





91/95


1 State the problem

2 Use the Inverse Function N ORMIN V to find the value ofz





92/95


1 State the problem


3 Unstandardize: Solve forx in the equation

z = x





93/95


1 State the problem



z = x

4 Example:

How tall does a young man need to be to be in the tallest 10%?





94/95


1 State the problem



z = x

4 Example:

How tall does a young man need to be to be in the tallest 10%?

What values should we use in N ORMIN V(p, , )?





95/95


1 State the problem



z = x

4 Example:How tall does a young man need to be to be in the tallest 10%?

What values should we use in N ORMIN V(p, , )?

z =N ORMIN V(0.9, 0, 1) = 1.2816, thenx = +z = 69 + 2.5

1.2816 = 72.2



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