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

Outline Chapter 5: Sampling Distributions Sampling distributions for counts and proportions (Section 5.1) Assignment # 3 --- solution 9/28/2019 Daniela Stan - CSC323

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

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

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

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

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

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

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

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

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

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

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

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: http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html 9/28/2019 Daniela Stan - CSC323