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

2. 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

3. 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.

4. 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.

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

6. 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

7. 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.

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

9. 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,…..

10. 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.

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

12. 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).

13. 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.

14. 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

15. 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.

16. Example
The standard deviation is :

17. 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

18. 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.

19. 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.

20. 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 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:

21. 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 .

22. 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

23. 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