Download Section 5.1 (part 2)

Bernoulli Trials
• Two Possible Outcomes
– Success, with probability p
– Failure, with probability q = 1  p
• Trials are independent.
Binomial Distribution
• For n Bernoulli trials, the number of
successes X is a binomial random variable.
The probability of k successes is given by
the binomial probability formula:
 n k
nk


P X  k     p 1  p 
k 
• As k varies with fixed n and p, the binomial
probabilities define a binomial probability
distribution over {0, 1, 2, …, n}.
Sampling Distribution of the
Count in a SRS
• When the population is much larger than the
sample, the count of X successes in a SRS of size
n has approximately the Binomial(n,p) distribution
(given that the true proportion of successes in the
population is p).
• As a rule of thumb, we use the binomial sampling
distribution for counts when the population is at
least 10 times as large as the sample.
Mean and Standard Deviation of a Binomial RV X
(i.e., of a sample count)
 X  np
 X  np1  p
Mean and Standard Deviation of a sample proportion, p̂
 pˆ  p
 pˆ 
p1  p 
n
Law of Large Numbers
• Informal: If n is large, the proportion of
successes in n Bernoulli trials will be very
close to p.
• Formal: For Bernoulli trials with n and p,
as n  ,
P pˆ  p     1
for all  > 0, where p̂ is the sample
proportion.
Binomial (n=100, p=1/2) Distribution
0.10
Probability
0.08
0.06
0.04
0.02
0.00
30
40
50
Number of Successes
60
70
Binomial (n=100, p=1/2) Distribution
With Normal Approximation Curve
0.10
Probability
0.08
0.06
0.04
0.02
0.00
30
40
50
Number of Successes
60
70
Normal Approximation
• Draw a SRS of size n from a large population having proportion p
of successes. Let X be the count of successes in the sample and
p̂ = X/n the sample proportion. When n is large, the sampling
distributions of the two statistics are approximately normal:

X is approximat ely normal N np, np1  p 

pˆ is approximat ely normal N  p,


p1  p  


n

• As a rule of thumb, we use the approximation for values of n and
p such that np  10 and n(1p)  10.
Example 1 – Normal Approximation of Counts
• Suppose you flip a balanced coin 1000 times. What is
the probability of getting between 480 and 532 heads?
Distribution of Number of Heads in 1000 Flips of a Balanced Coin
0.030
0.025
Probability
0.020
0.015
0.010
0.005
0.000
440
460
480
500
Number of Heads
520
540
560
Distribution of Number of Heads in 1000 Flips of a Balanced Coin
0.030
0.025
Probability
0.020
0.015
0.010
0.005
0.000
440
460
480
500
Number of Heads
520
540
560
Distribution of Number of Heads in 1000 Flips of a Balanced Coin
0.030
Approximate Normal Curve
0.025
Probability
0.020
0.015
0.010
0.005
0.000
440
460
480
500
Number of Heads
520
540
560
Distribution of Number of Heads in 1000 Flips of a Balanced Coin
0.030
Approximate Normal Curve
0.025
Probability
0.020
0.015
0.010
0.005
0.000
477 478 479 480 481 482 483
530
Number of Heads
531
532
533
534
535
Example 2 – Normal Approximation of Proportions
• A corporation receives 100 applications for a position
from recent college graduates in business. Assuming
that these applications constitute a random sample of
graduates in business, what is the probability that
between 25% and 35% of the applicants are women if
30% of all recent college graduates in business are
women?