6.2 Normal Distribution
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6.2/6.5: The Normal Distribution
The Normal Approximation to the Binomial
Distribution - Goals
•
•
•
•
•
Be able to sketch the normal distribution.
Be able to standardize a value.
Be able to use the Z-table.
Be able to calculate probabilities.
Be able to calculate percentiles (Inverse
calculations).
• Determine when you can use the normal
approximation to the binomial and perform
calculations using this approximation.
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Normal Distribution
𝑓 𝑥 =
1
(𝑥−𝜇)2
−
𝑒 2𝜎2
𝜎 2𝜋
where -∞ <  < ∞, σ > 0
X ~ N(,σ2)
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Shapes of Normal Density Curve
http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref
/process_simulations_sensitivity_analysis_and_error_analysis_modeling
/distributions_for_assigning_random_values.htm
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Graph of Normal Distribution
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Standard Normal Distribution (z curve)
𝑓 𝑥 =
1
(𝑥−𝜇)2
−
𝑒 2𝜎2
𝜎 2𝜋
where -∞ <  < ∞, σ > 0
𝑓 𝑧 =
1
2𝜋
μ = 0, 2 = 1
𝑧2
−
𝑒 2
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Cumulative z curve area
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Z-table
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Using the Z table
area right of z
=
area between z1 and z2 =
1

area left of z
area left of z1
–
area left of z2
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Procedure for Normal Distribution
Problems
1. Sketch the situation and shade the area to be
found. Optional but extremely useful.
2. Standardize X to state the problem in terms of Z.
3. Use Table III to find the area to the left of z.
4. Calculate the final answer.
5. Write your conclusion in the context of the
problem.
𝑋−𝜇
𝑍=
𝜎
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Percentiles
𝑋−𝜇
𝑍=
𝜎
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Symmetrically Located Areas
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Why Approximate the Binomial
Distribution?
1. Intervals
2. Computation (n large)
3. Inference
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Difficulties with the Normal
Approximation to the Binomial
1. Skewedness of the Binomial Distribution.
2. The Binomial Distribution is discrete.
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Continuity Correction
http://wiki.axlesoft.com/index.php?title=Continuity_correction
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Continuity Correction
Actual Value
P(X = a)
P(a < X)
P(a ≤ X)
P(X < b)
P(X ≤ b)
Approximate Value
P(a – 0.5 < X < a +0.5)
P(a + 0.5 < X)
P(a – 0.5 < X)
P(X < b – 0.5)
P(X < b + 0.5)
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