Download Standard Deviation of a Binomial Distribution

DG7 --- 10 minutes
Accel Math III
Unit #1: Data Analysis
Lesson #10: Calculating Mean,
Variance, and Standard
Deviation for a Binomial
Distribution
EQ: How do you calculate mean, variance,
and standard deviation of a binomial
distribution? How are these used to
interpret the data?
Recall: Create a probability distribution
where the random variable of
interest is number of heads
occurring when tossing 4 coins.
X = number of heads when tossing 4 coins
HINT: You can, but you don’t have to create a
sample space to complete this task.
Work smarter, not harder!!
How could you use Lists and binomialpdf to find
each value of P(xi) ?
0
0.0625
1
0.25
2
0.375
3
4
0.25 0.0625
Find the expected value for this task.
0(0.0625) + 1(0.25) +2(0.375)+3(0.25) +4(0.0625)
E(X) = __________________________
=2
Now calculate (n)(p)
______________
(4)(0.5) = 2
WOW!!
**Formula: Mean of a Binomial Distribution
(n)(p)
µ = __________________
Find the variance for this task.
σ2
2(0.0625) + (1 – 2)2(0.25) + (2 – 2)2(0.375)
(0
–
2)
= ____________________________
+ (3 – 2)2(0.25) + (4 – 2)2(0.0625) = 1
(4)(.5)(.5) = 1
Now calculate (n)(p)(q) ______________
WOW!!
**Formula: Variance of a Binomial Distribution
σ2
(n)(p)(q)
= ____________________________
Find the standard deviation for this task.
1 1
σ = _____________________
 ( 4)(.5)(.5)  1
**Formula: Standard Deviation of
a Binomial Distribution




n
p
q
σ = _____________________
Ex 1. A die is rolled 480 times. Find the
mean, variance, and standard deviation of
the number of 2’s that will be rolled.
X = # of twos occurring on 480 rolls of a die
Would
thistobe
a BINOMIAL
DISTRIBUTION
a this!!
I DO NOT
want
create
a probability
distributionorfor
NORMAL DISTRIBUTION?
WHY?
WHY?
µ
(n)(p) = (480)(1/6) = 80
= __________________
σ2
(n)(p)(q)
=
(480)(1/6)(5/6)
=
66.67
= ________________
σ
 1  5 
n  p q   480    8.2
= _____________________
 6  6 
Ex 2. The Statistical Bulletin published by
Metropolitan Life Insurance Co. reported that 2% of
all American births result in twins. If a random
sample of 8000 births is taken, find the mean,
variance, and standard deviation of the number of
births that would result in twins.
this be
a BINOMIAL
DISTRIBUTION
X =Would
# of twin
births
occurring
8000 birthsor a
NORMAL DISTRIBUTION?
WHY?
µ
= __________________
(n)(p) = (8000)(0.02) = 160
σ2
(n)(p)(q) = (8000)(0.02)(0.98) = 156.8
= ________________
σ
n  p q   80000.020.98  12.5
= _____________________
Get out a sheet of paper to use for Ex 3.
 Answer #6 & #7 first.
6.
a.
b.
c.
d.
Criteria needed to be Binomial Distribution:
Fixed # of trials n = 10 restaurants
Success --- unsatisfactory Failure --- satisfactory
Condition of one restaurant independent of others
p = 3/7 remains the same from restaurant to
restaurant
Yes, this scenario meets the criteria for
a binomial distribution.
7. NO, binomial distribution simply means there are only
two outcomes, NOT the probabilities are 50% and 50%.
 Now finish #1 - #5 on your own.
3
1. P( X  3)10 C3  
7
3
7
4
   0.1879
7
4
6
5
5
3 4
3 4
2. P( X  4)  P( X  5)10 C4     10 C5      0.4686
7 7
7 7
3. P( X  1)  P( X  1)  P( X  2)  ...  P( X  10)
or P( X  1)  1  P( X  0)
3
4." most likely"  E ( X )    (10)   4.3
7
Only because it’s a
Binomial Distribution
 3  4 
5. " variable"   (10)    2.44897
Only because it’s a
 7  7 
2
or
 3  4 
 7  7 
  (10)    1.5649
Binomial Distribution
Assignment:
Practice Worksheet: Calculating
Mean, Variance, and Standard
Deviation of Binomial Distributions
DG8 --- Tues