Transcript Sec. 7-6: The Normal Approximation to the

Sec. 7-6:
The Normal Approximation to
the Binomial Distribution
• We’ve solved binomial problems using
Table B—we now want to apply the
concept of the normal distribution curve to
solve binomial problems.
• We can make something called a
“continuity correction” to our binomial
information to solve it like a normal
distribution, as long as the following exists:
np ≥ 5 AND nq ≥ 5
1. Continuity Correction:
Add or subtract .5 onto/from the x value.
2. To solve binomial P(x)’s: CHECK np & nq
a. Determine the mean (μ) and the
standard deviation (σ) using the old
formulas:
μ = np and σ = √npq
b. Draw a picture if needed &
CONTINUITY CORRECT AS NEEDED.
c. Use the “original” z-score formula to
determine what value to look up to get
your probability from Table E.
z=x–μ
σ
** We are back to the original z-score
because we are trying to find the
probability from a ‘real life’ data value—not
a mean.
Here’s an example:
Northwest Airlines recently reported that its
on-time arrival rate is 72%. Find the
probability that among 80 randomly
selected flights:
a.) FEWER than 70 arrive on time.
b.) AT LEAST 70 arrive on time.
c.) EXACTLY 70 arrive on time.
First, identify your information: n = 80
p = .72 so…
q = .28
*** CHECK np ≥ 5 and nq ≥ 5
np = 57.6 and nq = 22.4 so we can do this problem
like a normal distribution.
** We need to use this info. to determine μ, & σ.
μ = (80)(.72) = 57.6
σ = √80(.72)(.28) = 4.0
Draw your picture: you need to C.C. to 69.5.
z = 69.5 – 57.6
4
z = 2.975  2.98
Look up 2.98 as usual, & because the shading
goes thru the mean so add .5.
P(x < 70 on time) = .9986
b) At least 70 arrive on time:
(you will need to C.C. to 69.5)
Z = 69.5 – 57.6 = 2.98
4
Look up 2.98 on the chart and SUBTRACT from .5
because the shading is a tail.
P(x ≥ 70 on time) = .0014
c) Exactly 70 arrive on time:
We will have to C.C. to BOTH sides of 70
because we’re trying to create a single value.
So we will have to do the z-score twice.
Z = 69.5 – 57.6
4
Z = 2.98
z = 70.5 – 57.6
4
z = 3.23
P(x = 70 on time) = .4999 - .4986 = .0013