1. Daniela Stan Raicu School of CTI, DePaul University
CSC 323 Quarter: Spring 02/03
Daniela Stan Raicu
School of CTI, DePaul University
9/28/2019
Daniela Stan - CSC323

2. Outline Chapter 5: Sampling Distributions
Sampling distributions for counts and proportions (Section 5.1)
Assignment # solution
9/28/2019
Daniela Stan - CSC323

3. Bernoulli Trials Examples:
A Bernoulli trial is a random event with only two possible outcomes: a “success” or “failure.”
Examples:
1. Flipping a fair coin is an example of a Bernoulli trial, since the outcome is either a “heads” or a “tails.”
In practice, it doesn’t matter which of these outcomes is called a “success” -- this in arbitrary, but must be
consistent throughout the course of solving of a given problem.
2. Whether a person selected at random from a population is a smoker or non-smoker can be viewed as a Bernoulli trial.
9/28/2019
Daniela Stan - CSC323

4. Binomial Random Variables
The total number X of “successes” observed in a series of n identical independent Bernoulli trials is a binomial random variable.
Examples of binomial random variables:
1. The total number of “heads” that occur after tossing a fair coin twice (X={0,1,2})
2. The number of people in a class of n students that are asthmatic.
3. The number of people in a group of n intravenous drug users who are HIV positive.
9/28/2019
Daniela Stan - CSC323

5. Binary variables
Notice that the data of the type we are discussing now are
categorical (nominal) with only two possible values (binary or
‘0/1’ data).
Statistics for
categorical data
9/28/2019
Daniela Stan - CSC323

6. Binomial Distributions
The probabilities associated with all possible outcomes of a binomial random variable form a binomial distribution.
Example: The binomial distribution for the number of heads following the flip of a fair coin twice is:
Pr(0 heads) = 0.25
Pr(1 heads) = 0.50
Pr(2 heads) = 0.25
The binomial distribution forms a family of distributions,
with each family member identified by two parameters.
The two parameters of a binomial distributions are:
n the number of independent trials
p the probability of success per trial
X ~ B(n, p), meaning
“X is a binomial random variable with parameters n and p.”
9/28/2019
Daniela Stan - CSC323

7. Binomial Mean & Standard Deviation
For example, X ~ b(2, 0.5) means “X is distributed as a binomial random variable with n = 2 and p = 0.5” (as would be seen in the tossing of a fair coin twice).
What is the mean and standard deviation of the count X?
Binomial Mean or Expected Value:
µx = n*p
Standard Deviation:
x = (n*p*(1-p))1/2
Apply the above concepts to: Problems 5.20, 5.22, 5.24/page 391
9/28/2019
Daniela Stan - CSC323

8. Sample Proportions Examples of dealing with sample proportions:
What proportions of a large lot of switches fail to meet specifications?
What percent of adults favor clothes shopping?
What percent of adults favor stronger laws restricting firearms?
In statistical sampling, the proportion p of “successes” in a population:
Count of successes in sample X
Size of sample = n
p = ^ ^
The proportion p does not have a binomial distribution; the sampling distribution of a sample proportion p is close to normal.
9/28/2019
Daniela Stan - CSC323

9. Sample Proportions ^
Let p be the sample proportion of successes in an SRS of size n drawn from a large population having population proportion p of successes.
The mean and standard deviation of the sample proportion are:
µp = p
p = (p*(1-p)/n)1/2
Unbiased estimator of
the population proportion p ^ ^
The variability of p about its
mean decreases as the sample
size decreases.
9/28/2019
Daniela Stan - CSC323

10. Normal Approximation for Counts and Proportions
Draw an SRS of size n from a very large population having population proportion p of successes. Let X be the count of successes in the sample and p =X/n the sample proportion.
When n is large, the distributions of these statistics are approximately normal:
X is approximately normal N (n*p , (n*p*(1-p))1/2)
p is approximately normal N (p, p*(1-p)/n)1/2)
How large is “large”?
n*p >=10 and n*(1-p)>=10
9/28/2019
Daniela Stan - CSC323

11. Normal Approximations for Counts and Proportions
Ex: A sample survey asked a nationwide random sample of 2500 adults if they agree or disagree that “I like buying new clothes, but shopping is often frustrating and time consuming”. Suppose that 60% of all adults would agree if asked this question. What is the probability that the sample proportion who agree is at least 58%?
Sample size n = 2500
Population proportion of answering “yes” p=.6
P (p>.58) = ? ^
9/28/2019
Daniela Stan - CSC323

12. Normal Approximations for Counts and Proportions
Sampling
distribution
µp = p=.6
p = (p*(1-p)/n)1/2 = .0098
P (p>=.58)=P(Z>=-2.04)=.9793 ^
9/28/2019
Daniela Stan - CSC323

13. The Sampling Distribution of a Sample Mean x
From previous lecture:
The sample mean x from a sample or an experiment is an estimate of the mean µ of the underlying population.
The sampling distribution of x is determined by:
The design used to produce the data
The sample size n
The population distribution.
Let x be the mean of an SRS of size n from a population
having mean µ and standard deviation
The mean and standard deviation of x are:
µx= µ x = /(n1/2) 
9/28/2019
Daniela Stan - CSC323

14. Recommended Problems Problems 5.28, 5.31, 5.33, 5.38, 5.40/page 403
Visit the site from below to see an example of normal
(or Gaussian) distribution obtained as an aggregation of
many smaller, but independent random events:
9/28/2019
Daniela Stan - CSC323