Download Chapter 5 - ficzkomath

Chapter 5.5
Normal Approximations to
Binomial Distributions
Sampling Distributions
In 4.2, we learned how to find binomial
probabilities.
For example, if a surgical procedure has an
85% chance of success and a doctor
performs the procedure on 10 patients, Find
the probability that exactly 2 are successful.
What if the surgical procedure on 150 patients
and you want to find the probability of
fewer than 100 successful surgeries?
I. Normal Approximation to a
Binomial Distribution

If np >5 and nq ≥5, then the binomial
random variable x is approximately
normally distributed with


mean, u = np
Std. Dev., σ
npq
Example 1 (p260)
1.
2.
Do TIY#1
II. Correction for Continuity

Binomial Distribution is discrete probability




To calculate exact binomial probabilities, use binomial
formula for each x value and then add them up
Geometrically, this corresponds to adding the areas of bars
in the probability histogram
Each bar has a width of one unit and x is the midpoint of the
interval.
To use a continuous normal distribution to
approximate a binomial probability, you must


Move 0.5 unit to the left and right of the midpoint (x-value)
to include all possible x-values in the interval.
This is called making a correction for continuity
Example 2 (p261)
1.
2.
3.
DO TIY#2
III.Guidelines
Using Normal Distribution to Approximate
Binomial Probabilities
1.
2.
3.
4.
5.
6.
1.
Verify that the binomial
distribution applies.
2.
Determine if you can use the
normal distribution to approximate
x, the binomial variable.
Find the mean u and standard
3.
deviation o for the distribution.
Apply the appropriate continuity
4.
correction. Shade the
corresponding area under the
normal curve.
Find the corresponding z-score(s). 5.
Find the probability.
6.
Specify n, p and q.
Is np > 5?
Is nq > 5?
u = np
o = npq
Add or subtract
0.5 from endpoints.
z=x–u
o
Use normalcdf( ?,? )
Example 3 (p262)

DO TIY#3
Example 4 (p263)

DO TIY#4
Example 5 (p264)

DO TIY#5
Homework

P265 – 267 #2-8, 20