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Advanced Business Statistics

▪ Continuous Probability Distributions

▪ Normal Distribution

▪ Student t Distribution

▪ Data Collection and Sampling

Winter 2022

Agenda

❑ Review Distributions

❑ Normal Distribution

❑ t-Student Distribution

❑ Sampling and Data Collection

Random Variables

A random variable is a function or rule that assigns a

number to each outcome of an experiment.

Alternatively, the value of a random variable is a

numerical event.

Two Types of Random Variables:

- Discrete Random Variable

– one that takes on a countable number of values

– E.g. values on the roll of dice: 2, 3, 4, …, 12

- Continuous Random Variable

– one whose values are not discrete, not countable

– E.g. time (30.1 minutes? 30.10000001 minutes?)

Probability Distributions

A probability distribution is a table, formula, or graph that describes the

values of a random variable and the probability associated with these

values.

➢ Discrete (Binomial, Poisson, ...)

Discrete variable can take on a countable number of values.

➢ Continuous (Uniform, Normal, …)

Continuous is one whose values are uncountable and have an infinite

continuum of possible values.

An upper-case letter will represent the name of the random variable,

usually X. Its lower-case counterpart will represent the value of the

random variable.

The probability that the random variable X will equal x is → P(X = x)

Probability

Notation

0 ≤ 𝑃 𝑥 ≤ 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ෍

𝑎𝑙𝑙 𝑥𝑖

𝑃 𝑥 = 1𝑎𝑛𝑑

Probability Distributions

Examples of discrete variables:

❖ number of defective items produced during a week (possible values

0,1,2,...)

❖ result of the toss of a fair die (1,2,3,4,5,6)

❖ result of the flip of a coin (tails = 0, heads = 1)

❖ budget for a project when there are 3 alternatives ($25,000 , $40,000

and $50,000)

Example 1

The Statistical Abstract of the United States is published annually. It contains

a wide variety of information based on the census as well as other sources.

The objective is to provide information about a variety of different aspects of

the lives of the country’s residents. One of the questions asks households to

report the number of persons living in the household. The following table

summarizes the data. Develop the probability distribution of the random

variable defined as the number of persons per household.

1 2 3 4 5 6 7 or

more Total

31.1 38.6 18.8 16.2 7.2 2.7 1.4 116

Number of Persons

Number of Household

(Millions)

Probability Distributions

X P(X)

1 31.1

116 =0.268

2 38.6

116 =0.333

3 18.8

116 =0.162

4 16.2

116 =0.140

5 7.2

116 =0.062

6 2.7

116 =0.023

7 or

1.4

116 =0.012

P X ≤ 3 = 0.162 + 0.333 + 0.268 = 0.763

The probability that a household has 3 or less

persons:

P X = 6 = 0.023

The probability that a household has 6 persons:

P X ≥ 5 = 0.062 + 0.023 + 0.012 = 0.097

The probability that a household has 5 or more

persons:

In Class Activity

A survey of Amazon.com shoppers reveals the following probability distribution

of the number of books purchased per hit.

a. What is the probability that an Amazon.com visitor will buy four books?

b. What is the probability that an Amazon.com visitor will buy eight books?

c. What is the probability that an Amazon.com visitor will not buy any books?

d. What is the probability that an Amazon.com visitor will buy at least one

book?

x 0 1 2 3 4 5 6 7

P(x) 0.35 0.25 0.20 0.08 0.06 0.03 0.02 0.01

In Class Activity

A university librarian produced the following probability distribution of the

number of times a student walks into the library over the period of a semester.

a. P( X ≥ 20 )

b. P( X = 60 )

c. P( X > 50 )

d. P( X > 100 )

x 0 5 10 15 20 25 30 40 50 75 100

P(x) 0.22 0.29 0.12 0.09 0.08 0.05 0.04 0.04 0.03 0.03 0.01

Continuous Distributions

Unlike a discrete random variable, a continuous random variable is one that

can assume an uncountable number of values.

❑ We cannot list the possible values because there is an infinite

number of them.

❑ Because there is an infinite number of values, the probability of each

individual value is virtually 0.

❑ Thus, we can determine the probability of a range of values only.

Probability Density Functions

A function f(x) is called a probability density function (over the range a ≤ x ≤ b

if it meets the following requirements:

• f(x) ≥ 0 for all x between a and b

• The total area under the curve between a and b is 1.0

f(x)

x ba

area=1

Normal Distribution

The normal distribution, with the well known “bell-shaped” curve, is defined by

its mean 𝜇 and its standard deviations σ. Its probability density function is

given by:

where 𝑒 = 2.71828…

𝜋 = 3.14159…

▪ Bell-shaped

▪ Symmetric about the mean 𝜇

▪ Total area under the curve is equal1

𝑓 𝑥 = 1

𝜎 2𝜋 𝑒 − 1 2 ( 𝑥−𝜇 𝜎

𝜇

𝜎

Normal Distribution

𝜇 = 5 𝜇 = 9 𝜇 = 13

Normal Distributions with the Same Variance but Different Means

Normal Distributions with the Same Means but Different

Variances

𝜎 = 8

𝜎 =13

𝜎 =16

Example 2

The daily hours spent on computer game of 100 students are shown in the

below table.

Student # hours Student # hours Student # hours Student # hours Student # hours

1 1.53 21 4.11 41 2.59 61 3.39 81 4.50

2 4.51 22 4.36 42 6.51 62 6.00 82 4.51

3 3.11 23 3.59 43 4.45 63 3.58 83 4.58

4 6.13 24 4.26 44 6.36 64 8.50 84 5.00

5 4.10 25 4.43 45 5.24 65 7.22 85 5.28

6 4.11 26 4.22 46 5.00 66 4.10 86 7.00

7 6.33 27 6.45 47 2.50 67 5.00 87 1.25

8 3.00 28 4.00 48 4.00 68 6.00 88 2.41

9 4.59 29 4.51 49 5.44 69 6.27 89 6.00

10 4.05 30 5.25 50 6.43 70 5.11 90 3.43

11 5.44 31 4.28 51 3.50 71 3.24 91 4.44

12 5.21 32 2.50 52 5.00 72 6.00 92 4.18

13 6.00 33 3.00 53 3.21 73 5.16 93 2.55

14 5.12 34 5.24 54 3.30 74 3.43 94 4.34

15 5.00 35 7.00 55 3.34 75 5.14 95 6.00

16 5.44 36 4.00 56 3.05 76 4.51 96 3.09

17 5.12 37 6.37 57 5.26 77 4.00 97 5.00

18 4.24 38 5.05 58 6.53 78 1.10 98 6.42

19 5.00 39 5.00 59 3.37 79 5.00 99 5.11

20 2.35 40 4.48 60 7.15 80 3.21 100 0.15

Normal Distribution

Hours Frequency

0 < X ≤ 1 1

1 < X ≤ 2 3

2 < X ≤ 3 6

3 < X ≤ 4 17

4 < X ≤ 5 27

5 < X ≤ 6 25

6 < X ≤ 7 16

7 < X ≤ 8 4

8 < X ≤ 9 1

100

0 1 2 3 4 5 6 7 8 9

Normal Distribution

• The probability of spending less than 3

hours on computer game:

• The probability of spending more than 6

hours on computer game:

• The probability of spending between 4

and 7 hours on computer game:

The Empirical Rule

The 68-95-99.7 Rule (the Empirical Rule)

In bell-shaped distributions, about 68% of the values fall within one

standard deviation of the mean, about 95% of the values fall within two

standard deviations of the mean, and about 99.7% of the values fall

within three standard deviations of the mean.

Standard Normal Distribution

To calculate the probability that a normal random variable falls into any

interval, the area in the interval under the curve must be computed. The

normal distribution function is not as simple as other distribution.

The random variable can be standardized by subtracting its mean µ and

dividing by its standard deviation σ and amount of the probability is

extracted from the normal distribution table.

When the variable is normal, the transformed variable is called a “standard

normal” random variable and denoted by Z whose μ = 0 and σ= 1; that is:

𝑧 = 𝑋 − 𝜇

𝜎

Standard Normal Distribution

P( Z ≤ -2.45 ) = 0.0071

P( Z ≤ -1.28 ) = 0.1003

0- 2.45

0- 1.28

Example 3

X is normally distributed with mean 100 and standard deviation 20. What is

the probability that X is less than 145?

𝑃 𝑋 < 145 = 𝑃 𝑋 − 100

20 < 145 − 100

20 = 𝑃 𝑍 < 2.25 = 0.9878

From normal

distribution table

Example 4

X is normally distributed with mean 1,000 and standard deviation 250. What is

the probability that

X lies between 800 and 1,100?

𝑃 800 < 𝑋 < 1100 = 𝑃 800 − 1000

250 < 𝑋 − 1000

250 < 1100 − 1000

250

= 𝑃 −0.8 < 𝑍 < 0.4 = 𝑃 𝑍 < 0.4 − 𝑃 𝑍 < −0.8

= 0.6554 − 0.2119 = 0.4435

From normal

distribution table

Example 5

The lifetimes of light bulbs that are advertised to last for 5,000 hours are

normally distributed with a mean of 5,100 hours and a standard deviation of 200

hours. What is the probability that a bulb lasts longer than the advertised figure?

𝑃 𝑋 > 5000 = 𝑃 𝑋 − 5100

200 > 5000 − 5100

200 = 𝑃 𝑍 > −0.5

= 1 − 𝑃 𝑍 < −0.5 = 1 − 0.3085 = 0.6915

From normal

distribution table

In Class Activity

a. P(Z < 2.23)

b. P(Z > 1.87)

c. P(1.04 < Z < 2.03)

In Class Activity

The long-distance calls made by the employees of a company are normally

distributed with a mean of 6.3 minutes and a standard deviation of 2.2

minutes. Find the probability that a call

a. lasts between 5 and 10 minutes.

b. lasts more than 7 minutes.

c. lasts less than 4 minutes.

Finding Values of Z

Often, we are asked to find some value of Z for a given probability, i.e.

given an area (A) under the curve, what is the corresponding value of

z (zA) on the horizontal axis that gives us this area? That is:

𝑃 𝑍 > 𝑍𝐴 = 𝐴

Finding Values of Z

Example 6: Because of the relatively high interest rates, most consumers

attempt to pay off their credit card bills promptly. However, this is not always

possible. An analysis of the amount of interest paid monthly by a bank’s Visa

cardholders reveals that the amount is normally distributed with a mean of $27

and a standard deviation of $7.

What interest payment is exceeded by only 20% of the bank’s Visa

cardholders?

𝜇 = 27 𝜎 = 7

𝑃 𝑍 > 𝑧𝐴 = 0.2 → 1 − 𝑃 𝑍 > 𝑧𝐴 = 0.2 → 𝑃 𝑍 < 𝑧𝐴 = 0.8 𝑧𝐴 = 0.84

𝑧 = 𝑋 − 𝜇

𝜎 → 0.84 =

𝑋 − 27

7 → 𝑋 = 32.88

Interest payment is exceeded by only 20% of the

bank’s Visa cardholders is $32.88.

Student’s t Distribution

In probability and statistics, Student's t-distribution (or simply the t-distribution)

is any member of a family of continuous probability distributions that arises

when estimating the mean of a normally distributed population in situations

where the sample size is small and the population standard deviation is

unknown.

The density function for the Student t distribution is as follows

𝑓 𝑡 = Γ[ 𝜈 + 1 2

]

𝜐𝜋Γ( 𝜈 2 ) 1 +

𝑡2

𝜈

−( 𝜈 + 1

2 )

𝜈 (nu) is called the degrees of freedom

Γ (Gamma function) is Γ(k)=(k-1)(k-2)…(2)(1)

Student’s t Distribution

In much the same way that µ and σ define the normal distribution, ν, the

degrees of freedom, defines the Student t Distribution:

As the number of degrees of freedom increases, the t distribution approaches

the standard normal distribution.

Determining Student t Values

The student t distribution is used extensively in statistical inference.

T-distribution Table in appendix lists values of .

That is, values of a Student t random variable with degrees of freedom

such that:

𝑡𝐴,𝜐

𝜐

𝑃(𝑡 > 𝑡𝐴,𝜐) = 𝐴

The values for A are pre-determined “critical”

values, typically in the 10%, 5%, 2.5%, 1%

and 0.5% range.

Determining Student t Values

For example, if we want the value of t with 10 degrees of freedom such that

the area under the Student t curve is 0.05:

t0.05,10=1.812

In Class Activity

Use the t table to find the following values of t.

𝑡0.10,15

𝑡0.10,23

𝑡0.025,83

𝑡0.05,195

Data, Statistics, and Information

Statistics is a tool for converting data into information:

• But where then does data come from?

• How is it gathered?

• How do we ensure its accurate?

• Is the data reliable?

• Is it representative of the population from which it was drawn?

Data Statistics Information

Methods of Collecting Data

There are many methods used to collect or obtain data for statistical

analysis. Three of the most popular methods are:

Surveys Direct Observation (E.g. Number

of customers entering a bank per hour)

Experiments (E.g. new ways to

produce things to minimize costs)

Questionnaire Design – Key Principles

1. Keep the questionnaire as short as possible.

2. Ask short, simple, and clearly worded questions.

3. Start with demographic questions to help respondents get started

comfortably.

4. Use dichotomous (yes/no) and multiple-choice questions.

5. Use open-ended questions cautiously.

6. Avoid using leading-questions.

7. Pretest a questionnaire on a small number of people.

8. Think about the way you intend to use the collected data when

preparing the questionnaire.

Simple Random Sampling

A simple random sample is a sample selected

in such a way that every possible sample of

the same size is equally likely to be chosen.

For example, drawing three names from a hat

containing all the names of the students in the

class is an example of a simple random

sample: any group of three names is as

equally likely as picking any other group of

three names.

Sampling

Sampling is a sub-set of a whole population. Sampling is often done for two

reasons:

• Cost → it’s less expensive to sample 1,000 television viewers

than 100 million TV viewers

• Practicality → performing a crash test on every automobile

produced is impractical

Three Sampling Methods

• Simple random sampling

• Stratified random sampling

• Cluster sampling

Stratified Random Sampling

A stratified random sample is obtained by separating the population into

mutually exclusive sets, or strata, and then drawing simple random samples

from each stratum.

Strata 1 : Gender

Male

Female

Strata 2 : Age

less than 20

20-30

31-40

41-50

51-60

More than 60

Strata 3 : Occupation

professional

clerical

blue collar

other

Cluster Random Sampling

A cluster sample is a simple random

sample of groups or clusters of elements

(vs. a simple random sample of individual

objects).

This method is useful when it is difficult or

costly to develop a complete list of the

population members or when the

population elements are widely dispersed

geographically.

Two major types of error can arise when a sample of observations is taken

from a population:

Sampling Error refers to differences between the sample and the population

that exist only because of the observations that happened to be selected for

the sample.

Increasing the sample size will reduce this error.

Non-sampling Errors are more serious and are due to mistakes made in the

acquisition of data or due to the sample observations being selected

improperly. Three types of non-sampling errors:

Increasing the sample size will not reduce this type of error.

Sampling and Non-Sampling Errors

• Errors in data acquisition

• Nonresponse errors

• Selection bias

References

• Business Statistics in Practice: Second Canadian Edition, Bowerman,

O'Connell, et al. McGraw-Hill, Third Canadian Edition

• G. Keller (2017) Statistics for Management and Economics (Abbreviated),

11th Edition, South-Western (students can also use the 8th edition of the

same textbook).

• M. Middleton (1997) Data Analysis Using Microsoft Excel, Duxbury. (A good

reference for basic statistical work with Excel.)

Thank you