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BCOR 1020Business Statistics
Lecture 12 – February 26, 2008
Overview
• Chapter 7 – Continuous Distributions– Continuous Variables– Describing a Continuous Distribution– Uniform Continuous Distribution– Normal Distribution– Standard Normal Distribution
Chapter 7 – Continuous Variables
• Discrete Variable – each value of X has its own probability P(X).
• Continuous Variable – events are intervals and probabilities are areas underneath smooth curves. A single point has no probability.
Events as Intervals:
Chapter 7 – Continuous Variables
• Probability Density Function (PDF) – For a continuous random variable, the PDF is an equation that shows the height of the curve f(x) at each possible value of X over the range of X.
PDFs and CDFs:
Normal PDF
Chapter 7 – Continuous Variables
• Continuous PDF’s:• Denoted f(x)• Must be nonnegative• Total area under
curve = 1• Mean, variance and
shape depend onthe PDF parameters
• Reveals the shape of the distribution
PDFs and CDFs:
Normal PDF
Chapter 7 – Continuous Variables
• Continuous Cumulative Distribution Functions (CDF’s):• Denoted F(x)• Shows P(X < x), the
cumulative proportion of scores
• Useful for finding probabilities
PDFs and CDFs:
Normal CDF
Chapter 7 – Continuous Variables
• Continuous probability functions are smooth curves.• Unlike discrete
distributions, the area at any single point = 0.
• The entire area under any PDF must be 1.
• Mean is the balancepoint of the distribution.
Probabilities as Areas:
b
adxxfbXaPbXaP )()()(
Chapter 7 – Continuous Variables
Expected Value and Variance:
Chapter 7 – Normal Distribution
Characteristics of the Normal Distribution:• Normal or Gaussian distribution was named for German
mathematician Karl Gauss (1777 – 1855).• Denoted N(, )• “Bell-shaped” Distribution• Domain is – < X < + • Defined by two parameters, and • Symmetric about x = • Almost all area under the normal curve is included in the
range – 3 < X < + 3 (Recall the Empirical rule.)
Chapter 7 – Normal Distribution
When does a random variable have a Normal distribution?• It is assumed in our experiment or problem.• Our variable is the sample average for a large sample.
(We will discuss why later.)• A normal random variable should:
• Be measured on a continuous scale.• Possess clear central tendency.• Have only one peak (unimodal).• Exhibit tapering tails.• Be symmetric about the mean (equal tails).
Chapter 7 – Normal Distribution
Characteristics of the Normal Distribution:
Chapter 7 – Normal Distribution
Characteristics of the Normal Distribution:• Normal PDF f(x) reaches a maximum at and has points
of inflection at +
Bell-shaped curve
Chapter 7 – Normal DistributionCharacteristics of the Normal Distribution:• All normal distributions have the same shape but differ in
the axis scales.
Diameters of golf balls
= 42.70mm = 0.01mm
CPA Exam Scores
= 70 = 10
We can define a standard normal distribution and a transformation to it in order to answer questions about any normal random variable!
Chapter 7 – Normal Distribution
Characteristics of the Standard Normal:• Since for every value of and , there is a different
normal distribution, we transform a normal random variable to a standard normal distribution with = 0 and = 1 using the formula:
z = x –
• Denoted N(0,1)
• Shift the point of symmetry to zero by subtracting from x.
• Divide by to scale the distribution to a normal with = 1.
Chapter 7 – Normal Distribution
Characteristics of the Standard Normal:• Standard normal PDF f(z) reaches a maximum at 0 and
has points of inflection at +1.
• Shape is unaffected by the transformation. It is still a bell-shaped curve.
• Entire area under the curve is unity.• A common scale from -3 to +3 is used.
• The probability of an event P(z1 < Z < z2) is a definite integral of…
• However, standard normal tables or Excel functions can be used to find the desired probabilities.
2
2
21)(
z
ezf
Chapter 7 – Normal DistributionCharacteristics of the Standard Normal:
• CDF values are tabled and we will use the N(0,1) tables to answer questions about all Normal variables.
Chapter 7 – Normal Distribution
Normal Areas from Appendices C-1 & C-2:• Appendix C-1 allows you to find the area under the curve
from 0 to z. (Draw on overhead)
• Appendix C-2 allows you to find all of the area under the curve left of z. (Hand-out)
• Using either of these tables, we can use symmetry and compliments to determine probabilities for the standard normal distribution.
Chapter 7 – Normal DistributionNormal Areas from Appendices C-1 & C-2: • Example: We can use this table to find P(Z < -1.96) and
P(Z < 1.96) directly.
P(Z < -1.96) = .025
P(Z < 1.96) = .975
Chapter 7 – Normal DistributionNormal Areas from Appendices C-1 & C-2: • Example: Having found P(Z < -1.96), we can use this
result, along with symmetry and the compliment to find several other probabilities…
.9500
P(Z < -1.96) = .025P(Z < 1.96) = 1 – P(Z < -1.96) = 1 - .025 = .975
P(-1.96 < Z < 1.96) = P(Z < 1.96) – P(Z < -1.96) = .975 - .025 = .950
Consider P(|Z| > 1.96) = 1 – P(|Z| < 1.96) = 1 – P(-1.96 < Z < 1.96) = 1 – .950 = .050
Clickers
Use the table from Appendix C-2 (hand-out or overhead) to determine P(Z < 2.10).
A = 0.0179
B = 0.1151
C = 0.4821
D = 0.8849
E = 0.9821
Clickers
Use the table from Appendix C-2 (hand-out or overhead) to determine P(Z < -1.20).
A = 0.0179
B = 0.1151
C = 0.4821
D = 0.8849
E = 0.9821
Clickers
Use the table from Appendix C-2 (hand-out or overhead) to determine P(-1.20 < Z < 2.10).
A = 0.0972
B = 0.1151
C = 0.8670
D = 0.8841
E = 0.9821