The normal approximation to the Binomial variable

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The normal approximation to the Binomial
variable
B(n,p)
N(µnp,s2np(1-p))
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MM example

• In a large bowl of MMs, the proportion of blues
  is 1/6 (or .17).
• X- the number of blue MMs in a sample of size 6
• XB(6, 1/6)
• Draw the probability histogram of X and compute
  its mean and SD

.5 .4 .3 .2 .1
0 1 2 3 4 5 6
Shape Skewed right Mean 6(1/6)1 SD
v6(1/6)(5/6).9
3

• Suppose we take a sample of size 30 from the MM
  bowl
• XB(30, 1/6)
• Describe the center, spread, and shape of the
  distribution of X

x P( X x ) 0.00
0.0037 1.00 0.0230 2.00
0.0682 . . . 30.00
0.000
Minitab ..\binomial in class.MPJ
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XB(30, 1/6)
Shape Smoother, more bell shaped Mean 30(1/6)5
SD v30(1/6)(5/6)2
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• Suppose we take a sample of size 90 from the MM
  bowl
• XB(90, 1/6)
• Describe the center, spread, and shape of the
  distribution of X

x P( X x ) 0.00 0.0000
1.00 0.0000 2.00 0.0000
3.00 0.0001 4.00 0.0002
. . 90.00
0.000
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XB(90, 1/6)
Shape Even smoother, more bell shaped very
close to a normal curve Mean 90(1/6)15
SD v90(1/6)(5/6)3.5
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• The binomial variable XB(90, 1/6) behaves
  approximately like a normal variable with mean 15
  and SD 3.5

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The Normal approximation to the Binomial
distribution

As sample size n gets large, the distribution of
the binomial random variable X is well
approximated by the normal distribution, with the
same mean and SD of the binomial variable If
XB(n,p) as n increases XgtN(µnp, svnp(1-p))
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The Normal approximation to the Binomial
distribution

• How large should n be?
• This depends on the value of p.
• If p is close to 0.5, then the normal
  approximation applies for small values of n
• If p is far from 0.5, larger values of n are
  needed.
• The following rule of thumb helps us decide when
  to use the normal approximation
• np15 and n(1-p) 15

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Example

• Check if the rule of thumb is satisfied for
• B(n6, p1/6)
• no, since np6(1/6)1lt15
• 2. B(n90, p1/6)
• yes, since np90(1/6)15 and
  n(1-p)90(5/6)75gt15

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Example of normal approximation to binomial

• You operate a restaurant. You read that sample
  survey by the National Restaurant association
  shows that 40 of adults are committed to eating
  nutritious food when eating away from home. To
  help plan your menu, you decide to conduct a
  sample survey in your own area. You will use
  random digit dialing to contact an SRS of 200
  households by telephone.
• If the national results hold in your area, it is
  reasonable to use B(200,0.4) distribution to
  describe the count X of respondents who seek
  nutritious food when eating out.
• What is the mean number of nutrition-conscious
  people in your sample if p.4 is true?
• Mean of B(200,.4)200(.4)80 SDv200(.4)(.6)v48
  6.93
• (b) Use the normal approximation to compute the
  probability that X lies between 75 and 85

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• Is the normal approximation appropriate?
• np200(.4)80gt15 n(1-p)200(.6)120gt18
• XgtN(80,6.932)

p(75X85)
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The normal approximation applies also for
proportions

• Next we will show that if XB(n,p),
• As n increases
• is approximately N(
  )

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The normal approximation applies also for
proportion
A poll that surveyed 500 people found that 45 of
them support military action in Iraq. X the
number of people that support military action in
Iraq in a sample of 500 people XB(500,0.45)
Is the normal approximation appropriate in this
case?

• Since np500(.45)225 and n(1-p)500(.55)275
• ? we can use the normal approximation
• µXnp225
• sXv np(1-p)v500(.45)(.55)v123.75 11.12
• XgtN(225, 11.122)

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• Lets define a new variable
• - the proportion of people that support
  military action in Iraq
• Instead of looking at the number of people who
  support that attack, we can look at the
  proportion of people who support the attack.
• Denote this proportion by
• How does behave?
• If X is Normal, any linear transformation of X
  is also normal.
• Since is a linear transformation
  of X -
• is approximately normal (note that X is
  approximately normal)
• Calculating the mean and SD of requires
  some definitions

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Mean and SD of a linear transformation of a
random variable

• If X is a random variable
• and a and b are fixed numbers
• µabXE(abX)abµX
• s2abXVar(abX)b2s2X

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Mean and variance of
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Summary - Normal approximation to the Binomial
distribution
If XB(n,p) as n increases (rule of thumb np15
and n(1-p) 15)
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Back to the example
µ p 0.4 s
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Example of normal approximation to binomial

• You operate a restaurant. You read that sample
  survey by the National Restaurant association
  shows that 40 of adults are committed to eating
  nutritious food when eating away from home. To
  help plan your menu, you decide to conduct a
  sample survey in your own area. You will use
  random digit dialing to contact an SRS of 200
  households by telephone.
• If the national results hold in your area, it is
  reasonable to use B(200,0.4) distribution to
  describe the count X of respondents who seek
  nutritious food when eating out.
• What is the mean number of nutrition-conscious
  people in your sample if p.4 is true?
• (b) Define by p-the proportion of people in the
  sample that seek nutritious food when eating out.
  pX/n.
• Use the normal approximation to compute the
  probability that the p is larger than 0.7.

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• XgtN(80,6.932)