Transcript a normal distribution.

Chapter S7
The normal distribution
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Learning Objectives
– Identify the properties of the normal distribution and normal
curve
– Identify the characteristics of the standard normal curve
– Understand examples of normally distributed data
– Read z-score tables and find areas under the normal curve
– Find the z-score given the area under the normal curve
– Compute proportions
– Check whether data follow a normal distribution
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Chapter S7
The normal distribution
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Learning Objectives continued...
– Understand and apply the Central Limit Theorem
– Solve business problems that can be represented by a
normal distribution
– Calculate estimates and their standard errors
– Calculate confidence intervals for the population mean
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Normal distribution
When the frequencies of observations for a large
population result in a frequency polygon that follows
the pattern of a smooth bell-shaped curve that
population is said to have a normal distribution.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Normal distribution
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bell-shaped
symmetrical about the mean
total area under curve = 1
approximately 68% of distribution is within
one standard deviation of the mean
approximately 95% of distribution is within
two standard deviations of the mean
approximately 99.7% of distribution is
within 3 standard deviations of the mean
Mean = Median = Mode
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Normal curves
same mean but different standard deviation
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Standard score (z-score)
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The z-score of a measurement is defined as the
number of standard deviations the measurement is
away from the mean. The formula is:
observed value - mean
Standard score  z 
standard deviation
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Standard score (z-score)
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If a distribution has a mean of μ and a standard
deviation of σ the corresponding z-score of an
observation is:
z
x-

© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Conversion to raw scores
Z-score is calculated to determine the
appropriate areas under any normal curve.
– To convert raw score of x (from a distribution with a
mean μ and standard deviation σ to a z-score,
subtract the mean from x and divide by the
standard deviation.
– To convert a z-score to a raw score x, multiply the
z-score by the standard deviation and add this
product to the mean. In equation form:
x    Ζ
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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The Central Limit Theorem
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If random samples of size n are selected from
a population with a mean μ and a standard
deviation σ , the means of the samples are
approximately normally distributed with a
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mean μ and a standard deviation n ,even if
the population itself is not normally distributed,
provided that n is not too small. The
approximation becomes more and more
accurate as the sample size n is increased.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Confidence intervals
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Point estimates
– a single estimate of an unknown population mean
can be obtained from a random sample
– different random samples give different values of
the mean
– a single estimate is referred to as a point estimate
– accuracy depends on:
• variability of data in the population
• size of the random sample
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Standard error of mean
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Standard error of the mean provides the
precise measure of accuracy of a point
estimate of the mean
Standard error of the mean 
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n
Where:
σ = population standard deviation
n = size of random sample
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Confidence intervals
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A range of values in which a particular
value may lie is a confidence interval.
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The probability that a particular value
lies within this interval is called a level
of confidence.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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