12.5 Working with Samples
12.6 Binomial Distributions
There are admirable potentialities in every human being.
Believe in your strength and your youth. Learn to repeat
endlessly to yourself, 'It all depends on me.'
Samples and Populations
Sample – gathers info from only part of a population.
x
Sample Proportion – the ratio , where x is the number of times an
event occurs in sample size n. n
Random Sample – all members of a population are equally likely to
be chosen (so this is a good representation of the population)
Ex1) In a sample of 500 teenagers, 328 had never attended a popular
music concert. Find the sample proportion for those who have never
attended a concert. Write the answer as a percent.
Bias
The Sunnyvale High School student council dance committee is trying to
decide whether to have a band or a DJ for the fall dance. They decided
that each of the three committee members should survey the students in
their homeroom classes. Identify any bias in this sampling method.
1) This is a “convenience” sample that is convenient for the committee
members.
2) Three homerooms may not accurately reflect the opinions of the
entire school, because three homerooms is probably a low
percentage of all the school’s homerooms.
3) This sampling method has a bias and is not random.
Margin of Error
When a random sample of size n is taken from a large population, the
sample proportion has a margin of error of about
1
***The larger the sample, the smaller the margin of error!

n
We use margin of error to give us an interval that is likely to contain
the true population proportion
Ex2) An opinion poll about the popularity of the mayor has a margin of
error of ±5%. Estimate the number of people who were surveyed.
Margin of Error
Ex3) A survey of 528 high school seniors found that 343 already had
career plans after high school.
a) Find the margin of error for the sample.
b) Use the margin of error to find an interval that is likely to contain the
true population proportion.
Binomial Experiments
Binomial Experiments:
-repeated trials
-each trial has 2 possible outcomes (success or failure)
-trials are independent (probability of success is constant throughout)
# of successes
# of failures
r n r
C
p
q
n r
trials
probability
of success
probability
of failure
Binomial Probability
Ex4) Brittany makes 90% of the free throws that
she attempts. Find the probability that she will
make exactly 6 out of 10 consecutive free throws.
Binomial Distribution
Ex5) One survey found that 80% of
respondents eat corn on the cob in circles
rather than from side to side. Assume that this
sample accurately represents the population.
What is the probability that, out of 4 people, at
least 2 of them eat corn on the cob in circles?
Pat least 2 successes   P2 successes   P3 successes   P4 successes 
12.5 Working with Samples
12.6 Binomial Distributions
12.5 #1-6, 9-11, 15-19, 22, 24
12.6 #8, 10, 15-20, 22
There are admirable potentialities in every human being.
Believe in your strength and your youth. Learn to repeat
endlessly to yourself, 'It all depends on me.'
Binomial Distribution
Ex 6) A fast food restaurant is attaching prize cards to every one of its
soft drink cups. The restaurant awards free drinks as prizes on three out
of four cards. Suppose you have three cards. Find the probability that
exactly one of these cards will reveal a prize using a tree diagram.
P (three prizes) = 1 3
3
3
P (two prizes) = 3
4
2
1
= 0.422
4
1
P (one prize)
3
=3 4
1
4
P (no prize)
=1 1
4
3
4
= 0.422
2
= 0.141
= 0.016
The probability that exactly one of three cards will reveal a free drink is
about 14%.