Transcript Document

Lesson 7 - 2
The Standard Normal Distribution
Quiz
• Homework Problem: Chapter 6 review
The Stanley Cup is a best of seven series to determine
the NHL champions. The following data represents the
number of games played, X, in the Stanley Cup to
determine a champion from 1939 to 2004
a) Determine the mean
X
Frequency
b) Determine the Std Dev
4
20
5
6
7
16
17
13
• Reading questions:
– Area to the right in a standard normal curve = ____________
– What methods are used to evaluate standard normal
probabilities?
Objectives
• Find the area under the standard normal curve
• Find Z-scores for a given area
• Interpret the area under the standard normal curve
as a probability
Vocabulary
• Zα – the Z-score that corresponds to the area under
the standard normal curve to the right of Zα is α
Properties of the Standard Normal Curve
• It is symmetric about its mean, μ = 0, and has a
standard deviation of σ = 1
• Because mean = median = mode, the highest point
occurs at μ = 0
• It has inflection points at μ – σ = -1 and μ + σ = 1
• Area under the curve = 1
• Area under the curve to the right of μ = 0 equals the
area under the curve to the left of μ, which equals ½
• As Z increases the graph approaches, but never
reaches 0 (like approaching an asymptote). As Z
decreases the graph approaches, but never reaches, 0.
• The Empirical Rule applies
Calculate the Area Under the Standard
Normal Curve
• There are several ways to calculate the area under the
standard normal curve
– What does not work – some kind of a simple formula
– We can use a table (such as Table IV on the inside back cover)
– We can use technology (a calculator or software)
• Using technology is preferred
• Three different area calculations
– Find the area to the left of
– Find the area to the right of
– Find the area between
Obtaining Area under Standard Normal Curve
Approach
Graphically
Solution
Shade the area to the left of za
Use Table IV to find the row and
column that correspond to za. The
area is the value where the row
and column intersect.
Find the area to the
left of za
P(Z < a)
Normcdf(-E99,a,0,1)
a
Shade the area to the right of za
Find the area to the
right of za
Use Table IV to find the area to the
left of za. The area to the right of za
is 1 – area to the left of za.
Normcdf(a,E99,0,1) or
1 – Normcdf(-E99,a,0,1)
P(Z > a) or
1 – P(Z < a)
a
Shade the area between za and zb
Find the area
between za and zb
Use Table IV to find the area to the
left of za and to the left of za. The
area between is areazb – areaza.
Normcdf(a,b,0,1)
P(a < Z < b)
a
b
Example 1
Determine the area under the standard
normal curve that lies to the left of
a
A. Z = -3.49
Normalcdf(-E99,-3.49) = 0.000242
B. Z = -1.99
Normalcdf(-E99,-1.99) = 0.023295
C. Z = 0.92
Normalcdf(-E99,0.92) = 0.821214
D. Z = 2.90
Normalcdf(-E99,2.90) = 0.998134
Example 2
Determine the area under the standard
normal curve that lies to the right of
a) Z = -3.49
Normalcdf(-3.49,E99) = 0.999758
b) Z = -0.55
Normalcdf(-0.55,E99) = 0.70884
c) Z = 2.23
Normalcdf(2.23,E99) = 0.012874
d) Z = 3.45
Normalcdf(3.45,E99) = 0.00028
a
Example 3
Find the indicated probability of the
standard normal random variable Z
a
a) P(-2.55 < Z < 2.55)
Normalcdf(-2.55,2.55) = 0.98923
b) P(-0.55 < Z < 0)
Normalcdf(-0.55,0) = 0.20884
c) P(-1.04 < Z < 2.76)
Normalcdf(-1.04,2.76) = 0.84794
b
Example 4
Find the Z-score such that the area under the
standard normal curve to the left is 0.1.
invNorm(0.1) = -1.282 = a
a
Find the Z-score such that the area under the
standard normal curve to the right is 0.35.
invNorm(1-0.35) = 0.385
a
Summary and Homework
• Summary
– Calculations for the standard normal curve can be
done using tables or using technology
– One can calculate the area under the standard normal
curve, to the left of or to the right of each Z-score
– One can calculate the Z-score so that the area to the
left of it or to the right of it is a certain value
– Areas and probabilities are two different
representations of the same concept
• Homework
– pg 381 – 383; 5 - 6, 9, 14, 21 – 22, 34, 37, 40