solved normal distri, z score

7/31/2019 Solved Normal Distri, Z Score

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The Normal Curveand Z-scores


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Properties of the NormalCurve:

Theoretical construction

Also called Bell Curve or Gaussian

Curve Perfectly symmetrical normal

distribution

The mean of a distribution is themidpoint of the curve its a unimodalcurve

The tails of the curve are infinite


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Properties

Standard normal curve has a mean =0 and standard deviation = 1.

General relationships: 1S = about68.26%

2S= about 95.44%

3S = about 99.72%


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Normal Curve


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Z-Scores

Are a way of determining the positionof a single score under the normalcurve.

Measured in standard deviationsrelative to the mean of the curve.

The Z-score can be used todetermine an area under the curveknown as a probability.


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Z-Scores

Z is a measure of how far away an observation (X)is from the mean in a normally distributed dataset.

Z = (X Mean) / Std DevA gets 64 marks in English, where mean is 50 & StdDev is 8.

z = (64 50) / 8 = 1.75

B gets 74 in German where mean is 58 and StdDev is 10

z = (74 58) / 10 = 1.6

B has higher marks but As performance is better.


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APPLICATIONS OF Z-SCORE

Questions that z score can answer:

Are you as good a student of French asyou are in Physics?

How many people did better or worsethan you on a test?

What percentage of people falls below agiven score?

What is the relative standing of a scorein one distribution versus another?


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Z-SCORE EXAMPLES

A student gets the same score of 60in two tests French and Physics

Why then is he happy with his score

in Physics but not French? Position relative to others in the

distribution is not the same


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PROPERTIES OF Z-SCORE

Calculate z for scores of

40 40-50/10 = -1

50 50-50/10 = 0

60 60-50/10 = 1

70 70-50/10 = 2

80 80-50/10 = 3 Z can be positive or negative

If an observation = mean, z=0

Z score tells us what proportion ofobservations are above or below that score


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Finding Probabilities

Areas under the curve can also beexpressed as probabilities.

Probabilities are proportions andrange from 0.00 to 1.00.

The higher the value, the greater theprobability (the more likely theevent). For instance, a .95 probability

of rain is higher than a .05 probability


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Example

After an exam, you learn that themean for the class is 60, with astandard deviation of 10. Suppose

your exam score is 70. What is yourZ-score?

Where, relative to the mean, does

your score lie? What is the probability associated

with your score (use Z table)?


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To solve:

Available information: Xi = 70

= 60

= 10

Formula:Z = (Xi ) /

= (70 60) /10

= +1.0


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- . . .mean (an area of 34.13% + 50% ) You are at the

84.13 percentile.


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What if your score is 72?

Calculate your Z-score.

What percentage of students have ascore below your score? Above?

What percentile are you at?


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Answer:

Z = 1.2

The area beyond Z = .1151(11.51% of marks are above yours)

Area between mean and Z = .3849 + .50 = .8849

(% of marks below = 88.49%)


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What if your mark is 55%?

Calculate your Z-score.

What percentage of students have ascore below your score? Above?

What percentile are you at?


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Answer:

Z = -.5

Area between the mean and Z = .1915 + .50 = .6915

(% of marks above = 69.15%)

The area beyond Z = .3085

(30.85% of the marks are below


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Another Question

What if you want to know how muchbetter or worse you did thansomeone else? Suppose you have

72% and your classmate has 55%?

How much better is your score?


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Answer:

Z for 72% = 1.2 or .3849 of areaabove mean

Z for 55% = -.5 or .1915 of areabelow mean

Area between Z = 1.2 and Z = -.5would be .3849 + .1915 = .5764


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Probability:

Lets say your classmate wont showyou the mark.

How can you make an informedguess about what your neighboursmark might be?

What is the probability that yourclassmate has a mark between 60%

(the mean) and 70% (1 s.d. above


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Answer:

Calculate Z for 70%......Z = 1.0

The area between the mean and Zis .3413

There is a .34 probability (or 34%chance) that your classmate has amark between 60% and 70%.


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The probability of your classmate having a markbetween 60 and 70% is .34 :

Example


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Example Family incomes in two particular neighborhoods

follow a normal distribution. For neighborhood

1 the mean is $26,500, with a standarddeviation of $15,000. For neighborhood 2, themean is $30,000 with a standard deviation of$18,000.

Q.1) What proportion of families inneighborhood 1 have incomes below the meanfamily income in neighborhood 2?

Q.2) What income level in neighborhood 1represents the top 10% for that neighborhood?

Q.3) What income level for neighborhood 2

Answer


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Answer Question : Family incomes in two particular

neighborhoods follow a normal distribution. Forneighborhood 1 the mean is $26,500, with a

standard deviation of $15,000. For neighborhood2, the mean is $30,000 with a standard deviation of$18,000.

Therefore: M1 = 26500, M2 = 30000 & STD 1 =15000, STD 2 = 18000.

1)What proportion of families in neighborhood 1have incomes below the mean family income in

neighborhood 2?

P(X


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Normal distribution curve

Z=01 2 3-1-2-30.233

0.0910Required region0.5+0.0910=0.5910

Q.1)

Answer


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Answer 2) What income level in neighborhood 1 represents

the top 10% for that neighborhood?

Top 10% means 90% must be below this incomefigure X. The Z score is given in this case , fromtables Z= 1.28

1.28 = (X - 26,500)/STD 1

Solving for x. X = $45,700.00

3) What income level for neighborhood 2represents the lowest 5% of incomes for thatneighborhoods?

Same as above. Z score for 5% lowest is -1.65


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Z=01 2 3-1-2-3

Top10%=0.1

Area=0.5-0.1=0.4Z=1.28

Q.2)


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Z=01 2 3-1-2-3

Lowest5%=0.05

Area=0.5-0.05 = 0.45Z= -1.65

Q.3)


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Answer

4) For neighborhood 1, whatpercentage of families have incomesthat fall between the mean incomes

of neighborhood 1 and neighborhood2?

P ( 26,500


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Answer by Using Excel

Calculations are also provided in the form ofan excel sheet (attached).

Attachment(s):Excel sheet

solved normal distri, z score

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