# 4 questions for introduction to statistic by using minitab 19 graphing

STATISTICS 1000Q

Winter Intersession 2020

Assignment 2

Professor Suman Majumdar

Print your name below

After you complete the assignment, save it under the filename

yourlastname2.

General Instructions

Answer the questions in the fields provided for and submit the resulting document through

HuskyCT.

Question Number

1a

1b

1c

1d

2a

2b

2c

3

4

Point Allotted

2

2

2

2

2

2

2

3

3

Point Scored

QUESTION 1

The shape of the graph of a binomial distribution depends on the value of both n and p. To

see how the shape changes for a fixed value of n, you will let p vary and graph each

probability distribution. Let X be a binomial random variable with n = 10.

a. For p = 0.11 obtain a bar chart of the binomial probability distribution.

b. For p = 0.50 obtain a bar chart of the binomial probability distribution.

c. For p = 0.89 obtain a bar chart of the binomial probability distribution.

d. Describe the effect of changing p.

QUESTION 2

Now see what happens when you hold p constant and vary n. Let X be a binomial random

variable with p = 0.25.

a. Obtain a bar chart of the binomial probability distribution for n = 5.

b. Obtain a bar chart of the binomial probability distribution for n = 50.

c. Describe the effect of changing n.

QUESTION 3

The location of a Normal distribution is determined by its mean , where as its shape is

determined by the standard deviation . To see the effect of changing , you are going to

graph two Normal probability density functions, one with = 100 and another with =

105, both having = 10. Recall that for each distribution the first value should be 3 = 30

below the mean, and the last value should be 3 above the mean. When MINITAB creates

the X values for you, for both distributions set the data IN STEPS OF 1. Overlay the two

density functions on the same graph (in MINITAB), and paste in the box below.

QUESTION 4

Now you want to see graphically the effect of changing the standard deviation of a Normal

distribution. Let = 100 for both distributions, but let = 10 for one and = 16 for the

other distribution. Recall that for each distribution the first value should be 3 below the

mean of 100, and the last value should be 3 above the mean of 100. When MINITAB

creates the X values for you, for both distributions this time set the data IN STEPS OF 2.

Overlay the two density functions on the same graph (in MINITAB), and paste in the box

below.

Chapter 4

Random

Variables and

Probability

Distributions

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 1

Content

1. Two Types of Random Variables

2. Probability Distributions for Discrete

Random Variables

3. The Binomial Distribution

4. Probability Distributions for Continuous

Random Variables

5. The Uniform Distribution

6. The Normal Distribution

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 2

Learning Objectives

1. Develop the notion of a random variable

2. Learn that numerical data are observed

values of either discrete or continuous

random variables

3. Present some simple discrete and continuous

random variables

4. Study two important families of random

variables and their probability models in

detail: the binomial and the normal families

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 3

Thinking Challenge

You’re taking a 33 question

multiple choice test. Each

question has 4 choices.

Clueless on 1 question, you

decide to guess. What’s the

chance you’ll get it right?

If you guessed on all 33

questions, what would be your

grade? Would you pass?

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 4

4.1

Two Types of Random

Variables

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 5

Discrete Random Variable

Random variables that can assume a

countable (finite or infinite) number of values

are called discrete.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 6

Discrete Random Variable

Examples

Experiment

Random

Variable

Possible

Values

Make 100 Sales Calls

# Sales

0, 1, 2, …, 100

Inspect 70 Radios

# Defective

0, 1, 2, …, 70

Answer 33 Questions

# Correct

0, 1, 2, …, 33

Count Cars at Toll

Between 11:00 & 1:00

# Cars

Arriving

0, 1, 2, …, ∞

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 7

Continuous

Random Variable

If the number of possible values of the

random variable is uncountable, then the

random variable is called continuous.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 8

Continuous Random Variable

Examples

Experiment

Random

Variable

Possible

Values

Weigh 100 People

Weight

45.1, 78, …

Measure Part Life

Hours

900, 875.9, …

Amount spent on food

$ amount

54.12, 42, …

Measure Time

Between Arrivals

Inter-Arrival 0, 1.3, 2.78, …

Time

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 9

4.2

Probability Distributions for

Discrete Random Variables

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 10

Discrete

Probability Distribution

The probability distribution of a discrete

random variable is a graph, table, or

formula that specifies the probability

associated with each possible value of the

random variable.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 11

Requirements for the

Probability Distribution of a

Discrete Random Variable X

1. p(x) ≥ 0 for all values of x

2. p(x) = 1

where the summation of p(x) is over all

possible values of X.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 12

Discrete Probability

Distribution Example

Experiment: Toss 2 coins. Count number of

tails.

Probability Distribution

Values, x Probabilities, p(x)

0

1/4 = 0.25

1

2/4 = 0.50

2

1/4 = 0.25

© 1984-1994 T/Maker Co.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 13

Visualizing Discrete

Probability Distributions

Listing

Table

{ (0, .25), (1, .50), (2, .25) }

# Tails

f(x)

Count

p(x)

0

1

2

1

2

1

.25

.50

.25

Graph

p(x)

.50

.25

.00

Formula

x

0

1

A L WA YS L E A R N I N G

2

p (x ) =

n!

px(1 – p)n – x

x!(n – x)!

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 14

Characterization of the

Probability Distribution of a

Discrete Random Variable X

If we have a two column table with the

columns labeled x and p(x), and the entries

in the p(x) column satisfy the two properties

1. p(x) ≥ 0 for all values of x

2. p(x) = 1,

then, the table in question represents the

probability distribution of a discrete random

variable X.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 15

Modifying Discrete Probability

Distribution Example

x_

_p(x)_

0

0.16

1

0.48

2

0.36

There exists a discrete random variable X

such that the table above represents the

distribution of X.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 16

Summary Measures

1. Expected Value (Mean of probability

distribution)

Weighted average of all possible values

= E(X) = x p(x)

2. Variance

Weighted average of squared deviation

about mean

2 = E[(X 2(x 2 p(x)

3.

Standard Deviation

● 2

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 17

Summary Measures

Calculation Table

x

p(x)

Total

x p(x)

x p(x)

A L WA YS L E A R N I N G

x–

(x – 2

(x – 2p(x)

(x 2 p(x)

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 18

Variance – Alternative Calculation

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 19

Variance – Alternative Calculation

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 20

Variance – Alternative Calculation

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 21

Thinking Challenge

You toss 2 coins. You’re

interested in the number

of tails. What are the

expected value,

variance, and standard

deviation of this random

variable, number of tails?

© 1984-1994 T/Maker Co.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 22

Expected Value & Variance

Solution*

x

p(x)

x p(x)

x–

0

.25

0

–1.00

1.00

.25

1

.50

.50

0

0

0

2

.25

.50

1.00

1.00

.25

= 1.0

(x – 2 (x – 2p(x)

2 .50

.71

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 23

Examples of Discrete Random Variables

The Experiment of Rolling a Die – Part 1

The Experiment of Rolling a Die – Part 2

The Minimum Problem

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 24

Probability Rules for Discrete

Random Variables

Let x be a discrete random variable with probability

distribution p(x), mean µ, and standard deviation .

Then, depending on the shape of p(x), the

following probability statements can be made:

Chebyshev’s Rule

Empirical Rule

P x x µ

0

.68

P x 2 x µ 2

34

.95

P x 3 x µ 3

89

1.00

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 25

Suggested Exercises

Work out the following exercises from the

Textbook :

4.12, 4.14, 4.15, 4.26, 4.32, 4.35, 4.39.

These exercises will not be collected or graded,

but let me know as questions arise.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 26

4.3

The Binomial Distribution

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 27

Binomial Distribution

Number of ‘successes’ in a sample of n

observations (trials)

Number of reds in 15 spins of roulette wheel

Number of defective items in a batch of 5 items

Number correct on a 33 question exam

Number of customers who purchase out of 100

customers who enter store (each customer is

equally likely to purchase)

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 28

Binomial Probability

Characteristics of a Binomial Experiment

1. The experiment consists of n identical trials.

2. There are only two possible outcomes on each trial.

We will denote one outcome by S (for success) and

the other by F (for failure).

3. The probability of S remains the same from trial to trial.

This probability is denoted by p, and the probability of

F is denoted by q. Note that q = 1 – p.

4. The trials are independent.

5. The binomial random variable x is the number of S’s in

n trials.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 29

Binomial Probability

Distribution

n x n x

n!

x

n x

p( x) p q

p (1 p )

x ! (n x)!

x

p(x) = Probability of x ‘Successes’

p = Probability of a ‘Success’ on a single trial

q = 1–p

n = Number of trials

x = Number of ‘Successes’ in n trials

(x = 0, 1, 2, …, n)

n – x = Number of failures in n trials

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 30

Binomial Probability

Distribution Example

Experiment: Toss 1 coin 5 times in a row. Note

number of tails. What’s the probability of 3

n!

tails?

x

n x

p( x)

p (1 p )

x !(n x)!

© 1984-1994 T/Maker Co.

5!

p (3)

.53 (1 .5)53

3!(5 3)!

.3125

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 31

Binomial Probability Table

(Portion)

n=5

p

k

.01

…

0.50

…

.99

0

.951

…

.031

…

.000

1

.999

…

.188

…

.000

2

1.000

…

.500

…

.000

3

1.000

…

.812

…

.001

4

1.000

…

.969

…

.049

Cumulative Probabilities

p(x ≤ 3) – p(x ≤ 2) = .812 – .500 = .312

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 32

Binomial Distribution

Characteristics

n = 5 p = 0.1

Mean

E(x) np

P(X)

1.0

.5

.0

X

Standard Deviation

npq

0

2

3

4

5

n = 5 p = 0.5

.6

.4

.2

.0

P(X)

X

0

A L WA YS L E A R N I N G

1

1

2

3

4

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

5

Slide – 33

Binomial Distribution

Thinking Challenge

You’re a telemarketer selling

service contracts for Macy’s.

You’ve sold 20 in your last 100

calls (p = .20). If you call 12

people tonight, what’s the

probability of

A. No sales?

B. Exactly 2 sales?

C. At most 2 sales?

D. At least 2 sales?

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 34

Binomial Distribution Solution*

n = 12, p = .20

A. p(0) = .0687

B. p(2) = .2835

C. p(at most 2) = p(0) + p(1) + p(2)

= .0687 + .2062 + .2835

= .5584

D. p(at least 2) = p(2) + p(3)…+ p(12)

= 1 – [p(0) + p(1)]

= 1 – .0687 – .2062

= .7251

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 35

Binomial probability calculations

using Minitab

Graphing a Binomial Distribution

Binomial Distribution Word Problems

Simulating from a Binomial Distribution

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 36

Suggested Exercises

Work out the following exercises from the

Textbook :

4.43, 4.47, 4.50, 4.52, 4.56, 4.59.

These exercises will not be collected or graded,

but let me know as questions arise.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 37

4.5

Probability Distributions for

Continuous Random Variables

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 38

Continuous Probability

Distribution

To describe the probability distribution for a discrete random

variable, we specified the possible values of the variable along

with the corresponding probabilities.

That approach will not work for a continuous random variable for

two related reasons. First, the number of possible values for a

continuous random variable is uncountable. Second,

if X is a continuous random variable,

then, for any number b,

𝑃 𝑋 = 𝑏 = 0.

A L WA YS L E A R N I N G

Copyright © 2018, 2014, and 2011 Pearson Education, Inc.

Slide – 39

Continuous Probability

Distribution

To describe the probability distribution for a continuous random

variable X, we have to specify 𝑃 𝑎 < 𝑋 < 𝑏 for every pair of
numbers a and b such that 𝑎 < 𝑏.
Since, for a continuous random variable X, 𝑃 𝑋 = 𝑏 = 0 for any
number b, we conclude
𝑃 𝑎

## We've got everything to become your favourite writing service

### Money back guarantee

Your money is safe. Even if we fail to satisfy your expectations, you can always request a refund and get your money back.

### Confidentiality

We don’t share your private information with anyone. What happens on our website stays on our website.

### Our service is legit

We provide you with a sample paper on the topic you need, and this kind of academic assistance is perfectly legitimate.

### Get a plagiarism-free paper

We check every paper with our plagiarism-detection software, so you get a unique paper written for your particular purposes.

### We can help with urgent tasks

Need a paper tomorrow? We can write it even while you’re sleeping. Place an order now and get your paper in 8 hours.

### Pay a fair price

Our prices depend on urgency. If you want a cheap essay, place your order in advance. Our prices start from $11 per page.