The Standard Normal Distribution Section 7'2

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The Standard Normal DistributionSection 7.2

• Alan Craig
• 770-274-5242
• acraig_at_gpc.edu

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Objectives 7.2

• Find the area under the standard normal curve
• Find z-scores for the given areas
• Interpret the area under the standard normal
  curve as a probability

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Properties of the Standard Normal Curve

• Symmetric about the mean, m 0
• Highest point occurs at m 0
• Inflection points at -1 and 1
• Area under the curve 1
• Area under the curve to right of 0 equals area
  under the curve to the left of 0 equals ½
• As z approaches 8, the graph approaches the
  x-axis
• The Empirical Rule (68, 95, 99.7) applies

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Finding Areasfrom Z-scores

• Three possibilities left, middle, right

z0
z1
z0
z0
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Finding Areasfrom Z-scores
z0
z1
z0
z0
Find area to right of Z z0 1 area left of z0
Find area to left of Z z0
Find area between Z z0 and Z z1 Area left of
z1 - area left of z0
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Finding Areasfrom Z-scores

• General approach
• Step 1 Draw the standard normal curve, label the
  z-score(s) of interest, and shade the appropriate
  region.
• Step 2 Use a table or calculator to find the
  area.

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Finding Areasfrom Z-scores Tables

• Find the area under the standard normal curve to
  the left of Z 1.34.

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Finding Areasfrom Z-scores Tables

• Find the area under the standard normal curve to
  the left of Z 1.34.
• First, sketch the graph

z01.34
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Finding Areasfrom Z-scores Tables

• See Table II front inside cover of the text and
  find the row partially shown below
• To find 1.34, find 1.3 in the column and .04 in
  the row.
• Z 0.00 0.01 0.02 0.03
  0.04 0.05
• 
  
• 1.3 0.9032 0.9049 0.9066 0.9082 0.9099
  0.9115
• The area to the left of z 1.34 is 0.9099.

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Finding Areasfrom Z-scores Tables

• Find the area under the standard normal curve to
  the right of Z -0.48.

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Finding Areasfrom Z-scores Tables

• Find the area under the standard normal curve to
  the right of Z -0.48.
• 1. Sketch the graph

z0 -0.48
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Finding Areasfrom Z-scores Tables

• Find the area under the standard normal curve to
  the right of Z -0.48.
• 2. Find area to left and subtract from 1
• Go to the intersection of the row with -0.4 and
  the column with .08 to find 0.3156.
• 1 0.3156 0.6844

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Finding Areasfrom Z-scores Tables

• Find the area under the standard normal curve to
  the between z0 -0.48 and z1 1.34.
• 1. sketch the graph

z0 z1 -0.48 1.34
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Finding Areasfrom Z-scores Tables

• 1. Find the area under the standard normal curve
  to the between Z -0.48 and Z 1.34.
• 2. Subtract the area to the left of the Z-score
• -0.48 from the area to the left of the Z-score
  1.34
• 0.9099 0.3156 0.5943

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Finding Z-scores from Given Areas

• To find z-scores for a given area, we reverse the
  process we have just done.
• Sketch the graph
• Find the area in the table closest to the given
  area. (If it falls exactly in the middle of two
  values, take the average of the two corresponding
  z-scores. See Ex 6 p. 291)
• Find the z-score by reading the hundredths digit
  from the top row and the ones and tenths digits
  from the leftmost column.

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Finding Z-scores from Given Areas

• Find the z-score such that the area under the
  standard normal curve to the left is 0.85
• Sketch the graph

Area 0.85
Z0 ?
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Finding Z-scores from Given Areas

• Find the z-score such that the area under the
  standard normal curve to the left is 0.85
• Find the value in the table closest to an area of
  0.85. That value is 0.8508 corresponding to a
  z-score of 1.04.

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Finding Z-scores from Given Middle Areas

• Find the z-scores that divide the middle 60 of
  the area under the standard normal curve from the
  area in the tails.

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Finding Z-scores from Given Middle Areas

• Find the z-scores that divide the middle 60 of
  the area under the standard normal curve from the
  area in the tails.
• Sketch the graph

Middle Area 0.60
Tail Area 0.20
Tail Area 0.20
Z0 ?
Z1 ?
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Finding Z-scores from Given Middle Areas

• Find the z-scores that divide the middle 60 of
  the area under the standard normal curve from the
  area in the tails.
• Note that if 60 is in the middle, then the
  remaining 40 is divided into 20 in each tail.
  Further, because the distribution is symmetric,
  z1 - z0. Thus, we find the z-score
  corresponding to 0.20 which is -0.84.
• z0 -0.84 , z1 0.84

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za Z-sub alpha

• za is the Z-score such that the area under the
  standard normal curve to the right of za is a.

Area is a
za
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za Z-sub alpha

• Find Z0.14
• Area to the right is 0.14, so the area to the
  left is 1- 0.14 0.86.
• From the table, the value is z 1.08

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Probability Notation

• P(a lt Z lt b) probability a standard normal random
  variable is between a and b
• P(Z gt a) probability a standard normal random
  variable is greater than a
• P(Z lt a) probability a standard normal random
  variable is less than a

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Example

• Find P(Z lt 0.93)
• Sketch the graph.
• Use Table II
• P(Z lt 0.93) 0.8238

0.93
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Questions

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