Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts, Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula

1. INTRODUCTION TO
STATISTICS & PROBABILITY
Chapter 5: Sampling Distributions
(Part 1)
Dr. Nahid Sultana
1

2. Chapter 5:
Sampling Distributions
5.1 The Sampling Distribution of a
Sample Mean
5.2 Sampling Distributions for Counts and
Proportions
2

3. 5.2 Sampling Distributions for Counts
and Proportions
3
 Binomial Distributions for Sample Counts
 Binomial Distributions in Statistical Sampling
 Binomial Mean and Standard Deviation
 Sample Proportions
 Normal Approximation for Counts and Proportions
 Binomial Formula

4. Study of sampling distributions with the simplest case of a random
variable, where there are only two possible outcomes.
Example: A sample survey asks 2000 college students whether they think
that parents put too much pressure on their children. We will view the
responses of these sampled students as an SRS from a population.
Here we have only two possible outcomes for a random variable.
Count and Sample proportion
Example: Here n = 2000 college
students and X=840 is the number
of students who think that parents
put too much pressure on their
children.
What is the sample proportion of
students surveyed who think that
parents put too much pressure on their
children? 0.42840/2000p ==

Let n = the sample size and
X = represent the r.v. that gives
the count for the outcome of interest.
When a random variable has two
possible outcomes, we can use the
sample proportion, , as a
summary.
X/np =

4

5. 5
Binomial distribution for sample
counts
The distribution of a count X depends on how the data are produced.
Here is a simple but common situation.
The Binomial Setting
1. There are a fixed number n of observations.
2. The n observations are all independent.
3. Each observation falls into one of just two categories, which for
convenience we call “success” and “failure.”
4. The probability of a success, call it p, is the same for each
observation.
A binomial setting arises when we perform several independent trials
(also called observations) of the same chance process and record the
number of times that a particular outcome occurs. There are four
conditions for a binomial setting:

6. 6
6
The count X of successes in a binomial setting has the binomial
distribution with parameters n and p, denoted by B(n, p).
where n is the number of trials of the chance process and p is the
probability of a success on any one trial. The possible values of X are
the whole numbers from 0 to n.
Note: Not all counts have binomial distributions;
Binomial Distribution
Example (Binomial Setting): Tossing a coin n times.
1. n is fixed;
2. each toss gives either heads or tails;
3. knowing the outcome of one toss does not change the probability
of an outcome on any other toss i.e. independent;
4. If we define heads as a success, then p=1/2 is the probability of
a head and is same for any toss.
The number of heads in n tosses is a binomial random variable X. The
probability distribution of X is called a binomial distribution.
Binomial distribution for sample counts (cont…)

7. 7
Binomial distribution for sample counts
(cont…)
Example (Binomial Distribution
Toss a fair coin 15 times. Give the distribution of X, the number of
heads that you observe.
We have 15 independent trials, each with probability of success
(heads) equal to 0.5.
So X has the B(15, 0.5) distribution.
Note: The binomial distributions are an important class of discrete
probability distributions.

8. 8
Binomial Distributions in Statistical
Sampling
The binomial distributions are important in statistics when we want to
make inferences about the proportion p of successes in a population.
Sampling Distribution of a Count
Choose an SRS of size n from a population with proportion p of
successes. When the population is much larger than the sample, the
count X of successes in the sample has approximately the binomial
distribution B(n, p).
Note: Usually we use the binomial sampling distribution for counts
when the population is at least 20 times as large as the sample.
The accuracy of this approximation improves as the size of the
population increases relative to the size of the sample.

9. 9
Binomial Mean and Standard
Deviation
If a count X has the binomial distribution based on n observations with
probability p of success, what is its mean and standard deviation?
Here are the facts:
If a count X has the binomial distribution with number of trials n
and probability of success p, the mean and standard deviation of
X are:
Mean and Standard Deviation of a Binomial Random Variable
)1( pnp
np
X
X
−=
=
σ
μ
Note: These formulas work ONLY for binomial distributions.
They can’t be used for other distributions!

10. 10
Normal Approximation for Binomial
Distributions
As n gets larger, something interesting happens to the shape of a
binomial distribution.
Suppose that X has the binomial distribution with n trials and success
probability p. When n is large, the distribution of X is approximately
Normal with mean and standard deviation
As a rule of thumb, we will use the Normal approximation when n is
so large that np ≥ 10 and n(1 – p) ≥ 10.
Normal Approximation for Binomial Distributions
µX = np

11. 11
Example:
11
A survey asked a nationwide random sample of 2500 adults if they
agreed or disagreed that “I like buying new clothes, but shopping is
often frustrating and time-consuming.”
Suppose that exactly 60% of all adult U.S. residents would say
“Agree” if asked the same question. Let X = the number in the sample
who agree. Estimate the probability that 1520 or more of the
sample agree.
49.24)40.0)(60.0(2500)1(
1500)60.0(2500
==−=
===
pnp
np
σ
μ
Check the conditions for using a Normal approximation.
Since np = 2500(0.60) = 1500 and n(1 – p) = 2500(0.40) = 1000 are
both at least 10, we may use the Normal approximation.
3) Calculate P(X ≥ 1520) using a Normal approximation.

12. 12
Sampling Distribution of a Sample
Proportion
As n increases, the sampling distribution becomes approximately
Normal.
Sampling Distribution of a Sample Proportion
.isondistributisamplingtheofThe pmean
:Thensuccesses.ofproportionsamplethebeˆLetsuccesses.of
proportionwithsizeofpopulationafromsizeofSRSanChoose
p
pNn
n
pp
p
)1(
isondistributisamplingtheofdeviationstandardThe ˆ
−
=σ
For large n, ˆp has approximately the N(p, p(1− p)/n distribution.
ˆp =
count of successes in sample
size of sample
=
X
n
sample.in thesuccesses""ofnumberthe
andˆproportionsampleebetween thconnectionimportantanisThere
X
p

13. 13
Binomial Formula
The number of ways of arranging k successes among n
observations is given by the binomial coefficient
for k = 0, 1, 2, …, n.
Note: n! = n(n – 1)(n – 2)•…•(3)(2)(1) ; and 0! = 1.
n
k





 =
n!
k!(n − k)!

14. 14
Binomial Probability
If X has the binomial distribution with n trials and probability p
of success on each trial, the possible values of X are 0, 1, 2,
…, n. If k is any one of these values,
P(X = k) =
n
k





pk
(1− p)n−k