Business Statistics 1
L1
Data and dataset:
 Data are the facts and figures collected, summarized, analyzed, and interpreted
 The data collected in a particular study are referred to as the data set
Elements, variables, and observations:
 The elements are the entities on which data are collected. Examples: individuals,
firms, countries
 A variable is a characteristics of interest for the elements. Example: shoe size, GDP,
number of employees
 A set of measurements
Scales of measurement:
Nominal scale:
 No natural ranking of categories; ex. Genders, names, colors
 The variable values can only be described in words, not numbers.
 = & ≠ to see if the categories are the same or not
Ordinal scale
 Natural ranking of categories
 E.g. grades, education level, the Likert scale
 Mathematical operations used: =, ≠, < and >
Interval scale
 Always numeric
 Natural ranking
 Variables have fixed measurement units
 Arbitrary zero point
 Mathematical operations used: =, ≠, <, >, + and Ratio scale
 Always numeric
 Natural ranking
 E.g. length, weight, age
 Variables have fixed measurement units
 Mathematical operations used: =, ≠, <, >, +, -, multi and div.
L2
Sample space (S) – Collection of all possible outcomes.
Example:
Toss a coin: S = {head, tail}
Roll a die: S = {1, 2, 3, 4, 5, 6}
Play a football game: S = {Win, draw, lose}
Sample point – An experimental outcome is called a sample point or an element.
Properties:
 One outcome in the sample space must occur.
 Two outcomes cannot occur at the same time
 Example: “Roll a die” experiment – When you roll a die it is only possible to get one
outcome.
Events – is a set of consisting of a specific collection of sample points
Compliment event (E):
Example:
S = All companies in Sweden
E = All manufacturing companies in Sweden
E^- = All non-manufacturing companies in Sweden
Intersection:The intersection between two events, A and B:
 Outcomes in both events A and B
 “AND” statement
 Denoted AnB
 Example:
- S = All students in Sweden
- A = All Female students in Sweden
- B = All students in Sweden owning a car
- A n B = All female students in Sweden owning a car
Union:
The union between two events, A and B:
 Outcomes in either events A or B or both
 “OR” statement
 Denoted: A u B
 Example:
- S = All cars produced by Volvo in 2020
- A = All cars in S with defective gearbox
- B = All cars in S with defective brakes
- A u B = All cars in S with defective gearbox or defective breaks or both
Mutually exclusive events:
Two events, A and B, are mutually exclusive if:
 The two events cannot occur simultaneously

I.e. A n B does not contain any sample points (written A n B = 0/)

Example:
- S = All animals in Sweden
- A = All cats in Sweden
- B = All dogs in Sweden
- A n B = All cats that are dogs in Sweden = 0/
- Cats cannot also be dogs. Thus, the events A and B are mutually exclusive events.
Probability of an event:
If S has a finite number of sample points and each outcome are equally as likely to happen,
then the probability of an event A is:
Example: Roll a die
 S = {1, 2, 3, 4, 5, 6}
 Roll a 4: A = {4}
 P(A) = n(A) / n(S) = 1/6
Probability of the complement of any even:
Example: Roll a die
 S = {1, 2, 3, 4, 5, 6}
 A = {4}

The probability that event A or B or both occurs is calculated as:
Example:
 A = the event that a students owns a bike
 B = the event that a students owns a car
Conditional probability:


The probability of an event given that another
event occurred is called conditional probability
Denoted and calculated as:
Bayes´ theorem
Combinations
- Combinations are arrangements without respect to order
Permutations
- Permutations are arrangements with respect to order
L3:
A random variable is a variable that associates a numerical value with each possible outcome
of an experiment. There’s two types of random variables:
 Continuous random variables
 Discrete random variables can assume a countable number of values, finite or infinite
Discrete random variables
Example:
Let X be the number of heads when tossing two coins. The possible outcomes are: HH, HT, TH
and TT
Expected value of discrete random variables:
Example:
If the probability distribution is known it is also possible to calculate the population variance
and standard deviation as:
And
Now when we have the p(x) value, and u-value, we can plot it into the calculation, Example:
The Bernoulli distribution:
 Take the value of 0 and 1 example: means like choosing YES/ NO- you are one
possible answer (winning or losing the football match.
 The formula is used when the equations only has two possible outcomes
Let X be distributed Bernoulli (Ber) with parameter π, denoted X – Ber(π)
 Expected value: E(X) = π
 Variance: V(X) = π(1-π)
 Example: (Coin toss)
Let X = 1, if the toss results in “head”
And X = 0, if the toss results in “tail”.
Then X – Ber(0.5)
Expected value: E(X) = π = 0.5
Variance: V(X) = π(1-π) = 0.5*(1-0.5) = 0.25
Binomial distribution:
L4
Continuous variable:
- Definition: A continuous random variable can assume any value in a particular
interval on the real line or in a collection of intervals
- Characteristics: There’s infinite possible values that a continuous random
variables assumes a singel specified value (e.g. P(X=x) = 0
 To get a higher/smaller probability you can increase/decrease the width of the
intervals
 Formula:
The normal distribution:
- A descriptive model that describes the real worlds situation
- Can take any values
The probability density function of the normal distribution
µ= the expected value of X = the mean
Sigma ^2 = standard deviation
Is:
Standard Normal Distribution:
To find the probability – you can look at the table on canvas to find the area you calculate.
There’s two ways to find the probability:
- Symmetry rule
Ex. P(Z>2.67) = P(Z<-2.67)
Awnser: 0.0038
- Complement rule
Ex: 1 – P(Z>2.67)
Awnser: 0.0038



P(Z < z)
- Z = Random variable
- z = number
Example: Find P(Z < 1.70)
- Awnser: 0.9554
Example 2: Find P(-2.13 < Z < 0)
- Awnser: P(-2.13 < Z < 0) = P(- infinity < Z < 0) – P(- infinity < Z < -2.13)
= 0.5 – 0.0166 = 0.4832
Standardization theorem:
If X in normally distributed with the expected value E(X) = miu and variance (sigma ^2)
Then:
Example:
Awnser:
The Inverse Transformation
Exercise
Ch4, Ex 19
Anwer:
I: p(A) 0,05+0,20=0,25
P(B) 0,20+0,25=0,45
P(C) 0,20+0,20+0,15=0,55
III
P(AandB) = empty because they do not have nothing in common
P(AandB)= P(empty)=0
IV:
If A n C= 0 or not ?
AnC= (E2 upphöjt i 2) = 0 not mutually exclusive
U=union
V:
B=S/B= (E1,E2,E5,E6,E7)
P(B)= 0,05+0,20+0,15+0,10+0,05=0,55
Extra exercises lecture 2 with answers.
I. P(regular U over 40)0 160+545+745+200/2000= 0,825
II. P(regular and Under 20)= 160/2000= 0,08