# Download The Normal Distribution as an Approximation to the Binomial

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```The Normal Distribution as an Approximation to the Binomial Distribution
For a large enough number of trials (n) the area under normal curve can be used to approximate
the probability of a binomial distribution.
Requirements:
 The number of trials is fixed.
 Each trial is independent of the others.
 There are only two outcomes, success (p) and failure (q).
 The probability of each outcome remains constant between trials.
 𝑛∙𝑝 ≥5
 𝑛∙𝑞 ≥5
To use the normal distribution to approximate the binomial, we must also define a mean (𝜇) and
standard deviation (𝜎).
𝜇 =𝑛∙𝑝
𝜎 = √𝑛 ∙ 𝑝 ∙ 𝑞
Continuity Correction Factor:
Since the binomial distribution is not continuous whereas the normal distribution is continuous,
we must apply the continuity correct factor to account for this. This is as simple as adding or
subtracting 0.5 from the discrete x-value in question.
For the discrete value:
 𝑥 <; 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 0.5 𝑎𝑛𝑑 𝑢𝑠𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑙𝑖𝑚𝑖𝑡 [𝑳𝑬𝑺𝑺 𝑻𝑯𝑨𝑵]
 𝑥 >; 𝑎𝑑𝑑 0.5 𝑎𝑛𝑑 𝑢𝑠𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑙𝑖𝑚𝑖𝑡 [𝑮𝑹𝑬𝑨𝑻𝑬𝑹 𝑻𝑯𝑨𝑵]
 𝑥 ≤; 𝑎𝑑𝑑 0.5 𝑎𝑛𝑑 𝑢𝑠𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑙𝑖𝑚𝑖𝑡 [𝑨𝑻 𝑴𝑶𝑺𝑻]
 𝑥 ≥; 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 0.5 𝑎𝑛𝑑 𝑢𝑠𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑙𝑖𝑚𝑖𝑡 [𝑨𝑻 𝑳𝑬𝑨𝑺𝑻]
 𝑥 =; 𝑎𝑑𝑑 𝒂𝒏𝒅 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 0.5 𝑎𝑛𝑑 𝑢𝑠𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝒂𝒏𝒅 𝑟𝑖𝑔ℎ𝑡 𝑙𝑖𝑚𝑖𝑡𝑠 [𝑬𝑿𝑨𝑪𝑻𝑳𝒀]
Middlesex Community College | Prepared by: Stephen McDonald
```
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