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Probability and Statistics
Normal Curves and Sampling Distributions
Chapter 6
Section 6
Normal Approximation to Binomial Distribution
and to Distribution
p̂
Essential Question: How can you use the normal distribution to approximate the values of
a binomial distribution?
Student Objectives: The student will state the assumptions needed to us the normal
approximation to the binomial distribution.
The student will compute the mean and standard deviation for the
normal approximation.
The student will use the continuity correction to convert a range of
r-values to the corresponding range of normal x values.
The student will convert the x-values to a range of standardized
z-scores and determine the desired probabilities.
The student will describe the sampling distribution for the
proportions p-hat.
Terms:
Continuity correction
Mean
Standard deviation
z-score
Key Concepts:
1.
2.
3.
The normal probability distribution is a good approximation of the binomial
distribution if: np > 5 and nq > 5.
The value of µ = np , where n is the number of events and p is the probability of the
event having a favorable outcome.
The value of the standard deviation, ! , is npq , n is the number of events, p is the
probability of a favorable outcome, and q is the probability of non-favorable outcome.
Remember: p + q = 1.
Equations:
z ! score :!!z =
x!µ
"
Normal Approximation to the Binomail Distribution
Consider a binomial distribution where
n = number of trials
r = number of successes
p = probability of success on a single trial
q
= 1! p = probability of failure on a single trial
If np > 5 and nq > 5, then r has a binomial distribution that is approximated by
a normal distribution with
µ = np
and
" = npq
Sampling Distribution for the Proportion p̂ =
r
n
Given
n = number of trials
r = number of successes
p = probability of success on a single trial
q
= 1! p = probability of failure on a single trial
If np > 5 and nq > 5, then the random variable p̂ can be approximated by
a normal random variable (x) with mean and standard deviation
µ p̂ = p
and
" p̂ =
pq
n
How To Perform Continuity Corrections
Convert the discrete random variable r (number of successes) to the
continuous normal random variable x by doing the following:
1. If r is a left point of an interval, subtract 0.5 to obtain the
corresponding normal variable x; that is x = r ! 0.5.
P (r > 6)
P (r " 7)
Examples:
P ( x " 6.5 )
P ( r " 13)
P ( x " 12.5 )
2. If r is a right point of an interval, add 0.5 to obtain the
corresponding normal variable x; that is x = r + 0.5.
P ( r < 12 )
P ( r # 11)
Examples:
P ( x # 11.5 )
P (r # 7)
P ( x " 7.5 )
3. If r1 is a left point and r2 is a right point of an interval, subtract 0.5
to obtain the corresponding normal variable x1; that is x1 = r1 ! 0.5
and add 0.5 to obtain the corresponding normal variable x 2 ; that is
x2 = r2 + 0.5.
P ( 4 < r < 15 )
Examples:
P ( 5 # r # 14 )
P ( 4.5 # x # 14.5 )
Graphing Calculator Skills:
None
P ( 3 # r # 17 )
P ( 2.5 # x # 17.5 )
P ( 24 # r < 32 )
P ( 24 # r # 31)
P ( 23.5 # x # 31.5 )
Sample Question:
1. It has been shown that 60% of leukemia patients who are given a completely comparable
bone marrow transplant will survive. If 500 patients are given a transplant:
a. What is the mean and standard deviation?
b. What is the probability that at least 325 patients will survive?
c. What is the probability that fewer than 280 survive?
d. What is the probability that between 290 and 310 (inclusive) survive?
e. What is the probability that more than 285 but less than 320 survive?
Homework Assignments:
Pages 314 - 317
Exercises: #1, 5, 9, 13, 17, and 21
Exercises: #3, 7, 11, 15, and 19
Exercises: #2, 6, 10, 14, and 18
Exercises: #4, 8, 12, 16, and 20