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Chapter 7
Normal Curves
and Sampling
Distributions
Understanding Basic Statistics
Fifth Edition
By Brase and Brase
Prepared by Jon Booze
The Normal Distribution
• A continuous distribution used for modeling
many natural phenomena.
• Sometimes called the Gaussian Distribution,
after Carl Gauss.
• The defining features of a Normal Distribution
are the mean, µ, and the standard deviation, σ.
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The Normal Curve
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Features of the Normal Curve
• Smooth line and symmetric around µ.
• Highest point directly above µ.
• The curve never touches the horizontal axis in
either direction.
• As σ increases, the curve spreads out.
• As σ decreases, the curve becomes more
peaked around µ.
• Inflection points at µ ± σ.
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Two Normal Curves
Both curves have the
same mean, µ = 6.
Curve A has a
standard
deviation of σ = 1.
Curve B has a
standard
deviation of σ = 3.
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Normal Probability
• The area under any normal curve will always
be 1.
• The portion of the area under the curve within a
given interval represents the probability that a
measurement will lie in that interval.
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The Empirical Rule
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The Empirical Rule
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The Empirical Rule
The masses of the adult ostriches at the Vilas
Zoo are normally distributed with a mean of 124
kg and a standard deviation of 10 kg.
What is the probability that a randomly-selected
ostrich will have a mass between 124 kg and
154 kg?
a). 0.15%
b). 99.85%
c). 49.85%
d). 47.5%
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The Empirical Rule
The masses of the adult ostriches at the Vilas
Zoo are normally distributed with a mean of 124
kg and a standard deviation of 10 kg.
What is the probability that a randomly-selected
ostrich will have a mass between 124 kg and
154 kg?
a). 0.15%
b). 99.85%
c). 49.85%
d). 47.5%
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Raw Scores and z Scores
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Raw Scores and z Scores
For a distribution with  = 10 and  = 2.5, find the
z score of the value x = 15.
a). z = 2
b). z = 12.5
c). z = 4
d). z = 3.16
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Raw Scores and z Scores
For a distribution with  = 10 and  = 2.5, find the
z score of the value x = 15.
a). z = 2
b). z = 12.5
c). z = 4
d). z = 3.16
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Distribution of z-Scores
• If the original x values are normally distributed,
so are the z scores of these x values.
– µ=0
– σ=1
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Using the Standard Normal Distribution
There are extensive tables for the Standard
Normal Distribution.
• We can determine probabilities for normal
distributions:
1) Transform the measurement to a z score.
2) Utilize Table 3 of the Appendix.
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Using the Standard Normal Table
• Table 3(a) gives the cumulative area for a given
z value.
• When calculating a z Score, round to 2 decimal
places.
• For a z Score less than –3.49, use 0.000 to
approximate the area.
• For a z Score greater than 3.49, use 1.000 to
approximate the area.
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Area to the Left of a Given z Value
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Area to the Right of a Given z Value
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Area Between Two z Values
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Using a z Table
Using Table 3 in the Appendix , find the probability
that z > 0.9.
a). 0.22
b). 0.09
c). 0.65
d). 0.18
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Using a z Table
Using Table 3 in the Appendix , find the probability
that z > 0.9.
a). 0.22
b). 0.09
c). 0.65
d). 0.18
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Normal Probability Final Remarks
• The probability that z equals a certain number is
always 0.
– P(z = a) = 0
• Therefore, < and ≤ can be used
interchangeably. Similarly, > and ≥ can be used
interchangeably.
– P(z < b) = P(z ≤ b)
– P(z > c) = P(z ≥ c)
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Inverse Normal Distribution
• Sometimes we need to find an x or z that
corresponds to a given area under the normal
curve.
– In Table 3, we look up an area and find the
corresponding z.
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Finding z Corresponding to a Given
Area A (0 < A < 1)
Left-tail case: the given area is to the left of z.
Look up the number A in the body of the table
and use the corresponding z value.
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Finding z Corresponding to a Given
Area A (0 < A < 1)
Right-tail case: the given area is to the right of z.
Look up the number 1 – A in the body of the
table and use the corresponding z value.
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Finding z Corresponding to a Given
Area A (0 < A < 1)
Center-tail case: the given area is symmetric and
centered above z = 0.
Look up the number (1 – A)/2 in the body of the
table and use the corresponding ±z value.
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Inverse Normal Distribution
Using Table 3 in the Appendix, find the range of z
scores, centered about the mean, that contain
70% of the probability.
a). –1.04 to 1.04
b). –2.17 to 2.17
c). –0.30 to 0.30
d). –0.52 to 0.52
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Inverse Normal Distribution
Using Table 3 in the Appendix, find the range of z
scores, centered about the mean, that contain
70% of the probability.
a). –1.04 to 1.04
b). –2.17 to 2.17
c). –0.30 to 0.30
d). –0.52 to 0.52
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Critical Thinking – How to tell if data
follow a normal distribution?
• Histogram – a normal distribution’s histogram
should be roughly bell-shaped.
• Outliers – a normal distribution should have no
more than one outlier
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Critical Thinking – How to tell if data
follow a normal distribution?
• Skewness –normal distributions are symmetric.
Use the Pearson’s index:
3( x  median)
Pearson’s index =
s
A Pearson’s index greater than 1 or less than
–1 indicates skewness.
• Normal quantile plot – using a statistical
software (see the Using Technology feature.)
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Terms, Statistics & Parameters
• Terms: Population, Sample, Parameter,
Statistics
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Why Sample?
• If time and resources are limited, we take
samples to learn about the population.
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Types of Inference
1) Estimation: We estimate the value of a
population parameter.
2) Testing: We formulate a decision about a
population parameter.
3) Regression: We make predictions about the
value of a statistical variable.
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Sampling Distributions
• To evaluate the reliability of our inference, we
need to know about the probability distribution
of the statistic we are using.
• Typically, we are interested in the sampling
distributions for sample means and sample
proportions.
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The Central Limit Theorem (Normal)
• If x is a random variable with a normal
distribution, mean = µ, and standard deviation =
σ, then the following holds for any sample size:
(n is the sample size)
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The Standard Error
• The standard error is just another name for the
standard deviation of the sampling distribution.
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The Central Limit Theorem
(Any Distribution)
• If a random variable has any distribution with
mean = µ and standard deviation = σ, the
sampling distribution of x will approach a
normal distribution with mean = µ and standard
deviation =  n as n increases without limit.
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Sample Size Considerations
• For the Central Limit Theorem (CLT) to be
applicable:
– If the x distribution is symmetric or
reasonably symmetric, n ≥ 30 should suffice.
– If the x distribution is highly skewed or
unusual, even larger sample sizes will be
required.
– If possible, make a graph to visualize how
the sampling distribution is behaving.
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Critical Thinking
• Bias – A sample statistic is unbiased if the
mean of its sampling distribution equals the
value of the parameter being estimated.
• Variability – The spread of the sampling
distribution indicates the variability of the
statistic.
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Normal Approximation to the Binomial
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Normal Approximation to the Binomial
A fair coin is flipped 200 times and the number
of heads, x, is counted.
Find the normal approximation of the standard
deviation for this experiment.
a). 50
b). 7.07
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c). 10
d). 100
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Normal Approximation to the Binomial
A fair coin is flipped 200 times and the number
of heads, x, is counted.
Find the normal approximation of the standard
deviation for this experiment.
a). 50
b). 7.07
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c). 10
d). 100
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Continuity Correction
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