Samples and Populations M248: Analyzing data Block A UNIT A4 Samples and Populations

UNIT A4: Samples and Populations Block A UNIT A4: Samples and Populations Contents Introduction Section 1: Choosing a probability model Section 2: The population mean Section 3: The population variance Terms to know and use Unit A4 Exercises

Section 1: Choosing a probability model The shape of a Bernoulli distribution: Bernoulli (p) The Bernoulli experiment is a probability experiment that satisfies these requirements: We have n=1 (One trial) The possible outcomes are: success (1) and failure (0) The probability of Success is p, the probability of Fail is q=1-p The variable of interest is the number of successes. X= 0 or 1 The range of X is 0,1.

Examples Let X be the random variable representing getting a raise let X be the random variable representing the number of heads in a single coin flip Let X be the random variable representing having a boy of a married couple.

Section 1: Choosing a probability model The shape of a Bernoulli distribution P= 1/3 P= 0.8 P= 1/2

Section 1: Choosing a probability model The shape of a binomial distribution: B(n,p) The binomial experiment is a probability experiment that satisfies these requirements: Each trial can have only two possible outcomes—success or failure. There must be a fixed number of trials. n The outcomes of each trial must be independent of each other. The probability of success must remain the same for each trial. p The probability of Fail must remain the same for each trial. q=1-p The variable of interest is the number of successes. X The range of X is 0,1,2,…..,n

Examples In a survey, 65% of the voters support a particular referendum. If 10 voters are chosen at random, let X be the random variable representing the number of voters who support the referendum. A certain large manufacturing facility produces 20,000 parts each week. The manager of the facility estimates that about 1% of the parts they make are defective. Let X be the random variable representing the number of defective parts.

Section 1: Choosing a probability model The shape of a binomial distribution B(8,0.5) B(10,0.25) B(7,0.8)

Section 1: Choosing a probability model The shape of a Geometric distribution: G(p) The Geometric experiment is a probability experiment that satisfies these requirements: Each trial can have only two possible outcomes—success or failure. The outcomes of each trial must be independent of each other. The probability of success must remain the same for each trial. p The probability of Fail must remain the same for each trial. q=1-p The variable of interest is the number of trials required to obtain the first success. X The range of X is 1,2,3,…..

Examples I have a spinner that has two colors: Red and Blue. The chance of landing on RED 0.7 and the chance on landing on BLUE is 0.3. Spin the spinner until you get a RED It is known that 20% of products on a production line are defective. Products are inspected until first defective is encountered.

Section 1: Choosing a probability model The shape of a Geometric distribution

Section 1: Choosing a probability model The shape of a normal distribution The following figures shows the probability density functions of the two typical members of the normal family The graphs of these p.d.f. are symmetrical about a single peak. These p.d.f. are typical: the graph of any normal p.d.f. has a single peak at about which it is symmetrical. The dispersion about this peak is determined by the parameter (the standard deviation).

Section 1: Choosing a probability model For a discrete probability distribution, the mode is the value of the highest probability of occurring, if there is just one such value. For a continuous probability distribution, a mode corresponds to a local maximum of the p.d.f. Since a p.d.f. may have more than one maximum point, a distribution can be multimodal; so it makes sense, for instance, to refer to a distribution with two maxima as bimodal.

Section 2: The population mean and Variance Discrete Random Variable Section 2.1: The mean for a discrete random variable Section 3.1: The variance of a discrete random variable For a discrete random variable X with probability mass function p(x), the (population) mean of X (or the expected value of X) is denoted by: The variance is : Or The standard deviation see example 2.1 and 2.2 and solve activity 2.1 & 2.2

Example The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five nights or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean.

Example The standard deviation is :

Sections 2 & 3: The population mean and variance Section 2.2: Families of distributions: the mean Section 3.2: Families of distributions: the variance Distribution Range of the random variable Mean Variance Activities & examples Bernoulli dist. X=0,1 Activity 2.3 page 150, Activity 3.2 page 160 Binomial dist. X=0,1,…,n Activity and example 2.4 page 151, Activity 3.3 page 161 Geometric dist. X=1,2,….. Activities 2.5, 2.6 and 2.7 page 152 and 153, Activity 3.4 page 162

Example : Bernoulli Consider the following game. You toss a coin. If you get the heads, you receive $1. If you get the tails, you receive none. Let X be the random variable for the payoff of this game. X has the Bernoulli distribution with success Probability 0.5.

Example : Binomial The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins.

Example: Geometric Distribution Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. What is the average no. of trials required to produce 1 success? What is the mean and variance of no. of trials required to produce 1 success Solution:

then the mean of X is the variance is Sections 2 & 3: The population mean and variance Section 2.3 & 3.3: The Mean and Variance of a normal distribution If the random variable X has a normal distribution with parameters that is then the mean of X is the variance is And the standard deviation of X is .

Terms to know and use Population mode Unimodal Bimodal and multimodal as applied to a probability distribution Population mean Expected value Expectation Population variance and population standard deviation

Unit A4 Exercises M248 Exercise Booklet Solve the following exercises: Exercise 16…………………………………….Page 8 Exercise 17 …………………………………....Page 8 Exercise 18 ……………………………………Page 8 Exercise 19 ……………………………………Page 9 Exercise 20 ……………………………………Page 9 Exercise 21…………………………………… Page 9