Mid-term Review

Business Statistics
MB - 1201

February 22, 2018

 Reviews
• Week 1: Intro to Statistics
• Week 2: Descriptive statistics
• Week 3: Basic probability
• Week 4: Discrete probability
• Week 5: Normal probability & sampling
distribution
• Week 6: Estimation

2
 Introduction to Statistics
What you should understand in this week:
1. Types of data
2. How to present data

3
Type of Data

4
table For Categorical Data
Type of Income
investment (in thousand Percenta
ge
US $)
Stock 46.5 39.1%
pie
Bond 32.0 26.9% chart
Certificate of 15.5 13.0%
deposit
Saving 25.0 21.0%
TOTAL 119.0 100.0%

Pareto
diagram
bar chart

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For Numerical Data

table
Class Frequen histogram
(range of
temperature) cy
10 < T ≤ 20 3
20 < T ≤ 30 6
30 < T ≤ 40 5
40 < T ≤ 50 4
50 < T ≤ 60 2

polygon

6
 Descriptive Statistics
What you should know in this week:

How to describe numerical data

7
 Numerical
Descriptive

Central
Quartiles Variation Shape
Tendency

Arithmetic
Viance Skewness
Mean

Geometric Standard
Boxplot
Mean deviation

Coefficient of
Median
variation

Mode Z- score

8
 Basic Probability
What you should understand in this topic:
1. Basic probability
2. Conditional probability
3. Bayes theorem

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 Remember!

“OR” is represented with “U“

“and” is represented with “ ∩”

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 Examples
The following Venn diagram presents results of the sample of households in
terms of purchase behavior for large screen HDTVs

A : Households planned to Among

  the taken samples,
purchase 1. How many households planned to
B : Households actually purchase large screen HDTV?
purchased
2. How many households actually purchased
large screen HDTV without planning?
S
A B 650 3. How many households planned or
purchased large screen HDTV?
5 20 10
0 0 0 4. How many households neither planned
nor purchased large screen HDTV?

5. How many households were taken as

𝑃( 𝐴 ′ ∩ 𝐵)  
𝑃( 𝐴  ∩ 𝐵′ ) samples?
𝑃( 𝐴 ∩ 𝐵)   𝑃( 𝐴 ′ ∩ 𝐵 ′)  
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 Conditional Probability
• Two events, A and B are independent if
and only if
𝑃(𝐴 𝐵)=P(A)

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 Bayes Theorem

The process of revising initial probability estimates in the light of new

information. This information could be obtained from market research,
scientific test, or consultants.

Bayes’ Theorem is used as normative tool of how we should revise our

probability assessment when new information is available.

𝑃(B|A)=𝑃(𝐴).𝑃(𝐴|𝐵)/𝑃(𝐴)

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 Example

In the past, 40% of the new-model televisions have been successful, and 60%
have been unsuccessful. Before introducing the new model, the marketing
research department conducts study and releases report.
In the past, 80% of the successful new-model televisions had received favorable
reports, and 30% of the unsuccessful new-model televisions had received
favorable reports.

For the new model of TV under consideration, marketing research department

has issued a favorable report. What is the probability that the television will be
successful?

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 Example (cont’d)
Given:
 
Probability of TV being successful,
Probability of TV being unsuccessful,
Probability of getting favorable report, given the TV was
successful,
Probability of getting favorable report, given the TV was
unsuccessful,

Question: Probability of the new TV being successful,

given that it has favorable report,

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 Example (cont’d)
You can also
utilize
decision
tree or table

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 Discrete probability distribution
What you should understand in this chapter:
1. Type of discrete probability distribution
2. Identify the type of discrete probability
distribution.
3. How to analyze each type:
1. Binomial
2. Poisson

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 Distinguishing
BINOMIAL and POISSON Distribution
BINOMIAL POISSON
• There is a fixed number of trial
 There is an continuous area of
(n)
opportunity (i.e. range of time,
• There are only 2 possible surface area, etc.)
outcomes (e.g. success or fail;
 You are interested in counting the
men or women; present or
number of times a particular
absent, etc.)
event occurs in that area.
• You are interested in one of the 2
 The average number is known ()
outcomes
• The probability of the outcome of
interest is known ()
• The probability () is always same
in every trial
For further and more detail explanation, read text-book page 181 and18 188
 Binomial

 
Probability distribution of the binomial random variable X, number of
event of interest in n independent trials, is:

 
Mean and variance of the binomial distribution:
and

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 Poisson
 

Probability distribution of the Poisson random variable X, representing the

number of outcomes occurring in a given area of opportunity :

 
Mean and variance of the Poisson Distribution:
and

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 Remember!

To save time, you can always use binomial and

poisson table!

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Normal probability and sampling distribution

What you should know about this chapter:

1. Calculate Z score and probability of Z score.
2. Read Z table.
3. Identify condition of normal and sampling
distribution.

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23
24
So what are the differences?

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 Confidence interval estimation
What you should understand:
1. What confidence interval means.
2. Utilize calculation in cases.
3. Utilize sample size calculation.

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Numerical Confidence Intervals Categorical
Data Data

Population Population
Mean Proportion

σ Known σ Unknown

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Numerical Determining Sample Categorical
Data Size Data

For the For the

Mean Proportion

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Problem Exercise
 Problem 1
• During 2017, “Ibak” shower gel sales has decreased. Therefore, in the beginning of
2018, the product manager started promotion coupon initiative to boost product
sales. From his observation in a convenience store in Bandung, he recorded that
7 out of 10 customers of the store bought “Ibak” shower gel. He also recorded that
among the customers who bought “Ibak” shower gel, 30% of them had promotion
coupon. Furthermore, among the customers who did not buy “Ibak” shower gel,
20% of them had promotion coupon. Based on this information, if selected at
random, determine:
a) What is the probability that the customer has coupon?
b) Given that the customer has coupon, what is the probability that he/she will
buy the product?

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 Problem 1
•
• During 2017, “Ibak” shower gel sales has decreased. Therefore, in the beginning of
 2018, the product manager started promotion coupon initiative to boost product
sales. From his observation in a convenience store in Bandung, he recorded that
7 out of 10 customers of the store bought “Ibak” shower gel. He also recorded that
among the customers who bought “Ibak” shower gel, 30% of them had promotion
coupon. Furthermore, among the customers who did not buy “Ibak” shower gel,
20% of them had promotion coupon. Based on this information, if selected at
random,

Given:

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 Solution
•   What is the probability that the customer has coupon?
a)

\ Probability that customer have promotion coupon is 0.27

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 Solution
•   What is the probability that the customer has coupon?
a)

\ Probability that customer have promotion coupon is 0.27 or 27%

b) Given that the customer has coupon, what is the probability that he/she will buy
the product?

 Given that the customer has coupon the probability that he/she will buy
the product is 0.78 or 78%

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 Problem 2
In 2016, Andy, an SBM student agreed to record his travels from his house to campus.
He has three alternative routes to reach campus from his house, namely route A, B,
and C. Andy was asked to record the congestion level he experienced each day he
traveled to campus in three levels: light traffic, moderate traffic, and heavy traffic.
At the end of the year, it was recorded that he had made 200 travels to campus
throughout the year. Among those, he chose route A for 50 times, route B for 80
times, and route C for 70 times.
Among his experiences in using route A, 40% of the times he had light traffic, 48% of
the times he had moderate traffic, and 12% of the times he had heavy traffic. Among
his experiences in using route B, 40% of the times he had light traffic, 48% of the times
he had moderate traffic, and 12% of the times he had heavy traffic. Finally, among his
experiences in using route C, 40% of the times he had light traffic, 48% of the time he
had moderate traffic, and 12% of the time he had heavy traffic.

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 Questions
a) What is the probability that Andy chose route A and had heavy traffic?
b) What is the probability that he experience heavy traffic?
c) Can we say that the Andy’s choice of route is independent from the
congestion level?
d) If now in February 2017, Andy reported that he had light traffic today,
what is the probability that he had used route A?

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 Answer
•  
a) What is the probability that Andy chose route A and had heavy traffic?

 
b) What is the probability that he experience heavy traffic?

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 Answer
•  
c) Can we say that the Andy’s choice of route is independent from the congestion
level?
while

 Andy’s choice of route is not independent from the congestion level. It is likely that
he choose his route based on information he get about the congestion level.
 
d) If now in February 2017, Andy reported that he had light traffic today, what is the
probability that he had used route A?

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 Problem 3
You are manager of a restaurant that opens only for dinner. You have 10
reservations for tonight. According to your experience, you know that
probability that customers actually come after doing reservation is 80%.
a) What is the probability that all customers who did reservation actually
come for dinner?
b) What is the probability that at least 1 customer who did reservation do
not come?
c) If there are three customers on waiting list, what is the probability that
all waiting-list customers will be served?

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 Binomial / Poisson?
Is there a fixed number of trial?

Is there only 2 possible outcome

Can you know probability of an outcome?

Is there an area of opportunity defined?

Can you know the average number?

BINOMIAL

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 Remember!

BINOMIAL POISSON

• There is a fixed number of trial (n)  There is an continuous area of

• There are only 2 possible opportunity (i.e. range of time,
outcomes (e.g. success or fail; surface area, etc.)
men or women; present or  You are interested in counting
absent, etc.) the number of times a particular
• You are interested in one of the 2 event occurs in that area.
outcomes
 The average number is known
• The probability of the outcome of ()
interest is known ()
• The probability () is always same
in every trial

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 Binomial Distribution
 
a) What is the probability that all
customers who did reservation actually
•customer
b) What is the probability that at least 1
  who did reservation do not
come for dinner? come?

Given: n=10,  = 0.8 Given: n=10,  = 0.2

P(X=10) = 0.1074
Probability that all 10 persons who did Atau dengan pendekatan lain:
reservation come is 10.74% Given: n=10,  = 0.8
 

Probability that at least 1 customer

who did reservation do not come is
89.26%

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•waiting-list
c) If there are three customers on waiting list, what is the probability that all
  customers will be served?

Given: n=10,  = 0.2

 Probability that all 3 waiting list customers will be served is 32.22%

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 Problem 4
You are a manager of a restaurant that opens only for dinner. You have 12
tables in your restaurant. However, according to your experience, you know
that on average you have 10 reservations for every night.
a) What is the probability that you have all tables reserved for a night?
b) What is the probability that you get less than 10 reservations for a night?
c)  Three of your tables are currently unavailable. What is the probability
that you should reject any reservation for a night?

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 Binomial / Poisson?
Is there a fixed number of trial?

Is there only 2 possible outcome

Can you know probability of an outcome?

Is there an area of opportunity defined?

Can you know the average number?

POISSON

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 Poisson

•  
a)
 Probability that all tables in the restaurant is reserved is 9.48%
 
Or P(X>12),

 Probability that all tables in the restaurant is reserved is 30.32%

b) a. 𝑃ሺ𝑋 < 10ሻ= 𝑃ሺ𝑋 = 0ሻ+ 𝑃ሺ𝑋 = 1ሻ+ 𝑃ሺ𝑋 = 2ሻ+ 𝑃ሺ𝑋 = 3ሻ+ 𝑃ሺ𝑋 = 4ሻ+ 𝑃ሺ𝑋 = 5ሻ+ 𝑃ሺ𝑋 = 6ሻ+
𝑃ሺ𝑋 = 7ሻ+ 𝑃ሺ𝑋 = 8ሻ+ 𝑃ሺ𝑋 = 9ሻ
𝑃ሺ𝑋 < 10ሻ = 0 + 0.0005 + 0.0023 + 0.0076 + 0.0189 + 0.0378 + 0.0631 + 0.0901 + 0.1126 + 0.1251
= 0.4580

 Probability that there is less than 10 reservation for a night is 45.8%

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•you should reject any reservation for a night?
c) Three of your tables are currently unavailable. What is the probability that

 Probability that you will reject any reservation if there are 3 tables
unavailable is 54.2%

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 Problem 5
Mobil99 will try to expand its business next month in South Jakarta’s strategic areas by
adding a dedicated division for handling used cars’ full engine restoration before they
sell them. Previously, Mobil99 assigned the engine restoration to a partnering
workshop in Bekasi, but because of the workshop’s disappointing performance in the
last couple of months the partnership will be terminated soon. From past experience
in its newest opened branch, 2 out of 10 used cars would need a full engine
restoration. As the general manager of Mobil99 South Jakarta area, you are eager to
investigate the engine restoration division’s workload forecast based on the
information stated above.
a) Suppose that there are 20 used cars in Mobil99’s South Jakarta branch, what is
the probability that none of the car will need full engine restoration?
b) What is the probability that less than 4 cars will need full engine restoration?

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 BINOMIAL

Given:
n = 20
p = 0.2

a. Suppose that there are 20 used cars in Mobil99’s South Jakarta branch, what is the
probability that none of the car will need full engine restoration?
P (X=0) = 0.011529215
from table we get, P(X=0) = 0.0115
Thus, the probability that none of the car will need full engine restoration is 0.0115 or 1.15%

b. What is the probability that less than 4 cars will need full engine restoration?
P (X<4) = P (X=0)+P(X=1)+P(X=2)+P(X=3)
= 0.057646075 + 0.136909429 + 0.205364143
= 0.4114489
So, the probability that less than 4 cars will need full engine restoration is 0.4114 or 41.14%

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 Problem 5 Cont’d
To analyze the situation further, you want to know the car capacity for the garage that
you need to build for the new division where they will do the cars’ engine restoration.
From the data that you have from past experiences with the partnering workshop in
Bekasi, you know that every 2 weeks there are 6 used cars that their engines are being
fully restored in the shop. A full engine restoration would take exactly one week for
each car remembering that the mechanics need to make sure that those cars are in
prime and “ready to sell” condition. Based on the information above, the chief of the
newly formed engine restoration division suggested that the garage would only need a
capacity of 3 cars. Just to make sure, you want to make a forecast of how many car(s)
will be in the garage in a week for a full engine restoration.
c) What is the probability that there will be less than 2 cars in the garage in one week
for a full engine restoration?
d) You decided that you will follow the chief’s suggestion to build the garage with 3
cars capacity if the probability of more than 3 cars will be in the garage in one
week is not more than 30%. Will you follow the chief’s suggestion? Explain your
answer using the cars per week forecast!

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 POISSON

Given:
e = 2.718
λ=3

c) What is the probability that there will be less than 2 cars in the garage in one week
for a full engine restoration?
P (X<2) = P (X=0)+P(X=1) = 0.1992
So, the probability that there will be less than 2 cars in the garage in one week is 0.1992
or 19.92%.

d) You decided that you will follow the chief’s suggestion to build the garage with 3
cars capacity if the probability of more than 3 cars will be in the garage in one week
is not more than 30%.
P (X>3) = 1 - P (x<3) = 1 - 0.647 = 0.353
So, I will not follow the chief's suggestion to build the garage with 3 cars capacity.

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 Problem 6
Rocket Sales Corporation (as in movie Rocket Singh: Salesman of the Year, 2009) has
embarked on a service quality improvement effort. One of its project is related to
maintaining the target upload speed for its Internet services subscribers. Upload
speed are measured on a standard scale in which the target value is 1.000. Data
collected over the past years, indicate that the upload speed is approximately
normally distributed, with median value of 1.017 and there is approximately 20%
cases where the upload speed is less than 0.933.

a) For evaluation, the upload speed is measured each day, at 25 random times. What
is the probability that the average upload speed is higher than 1.000?
b) For improvement, technical team proposes two alternative scenarios: (1)
utilization of technology A to increase the mean upload speed to 1.05 and (2)
utilization of technology B to reduce the standard deviation to 75% of current
value. Which scenario will be the best to maximize the probability of upload speed
above the target?

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 Solution
•  
The first step is to find the standard deviation of the population.
There is only 20% cases where the upload speed is less than 0.933. This
clause can be expressed mathematically as

From the table of cumulative normal distribution,

Thus,

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 Solution
•   For
a) this case, the sample size is . Thus, the probability of the average
speed exceed 1.000 can be calculated as follow:

Z-value for is

Then, the probability of mean value is

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 Solution
• The
b)   outcome of both scenarios •  
Technology B: and
can be evaluated as follow: Z-value for is

Probability of upload speeds exceed the

Technology A: and target
Z-value for is

Probability of upload speeds  

exceed the target So, technology B serves 58.97% chance
of exceeding upload speeds, while
technology A serves only 69.15%. Both
scenarios improve the performance of
Rocket Sales Corporation, but utilization
of technology A is the best alternative.
 

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 Problem 7
In 2015, dailysocial.id conducted a survey to know the payment method used most by
Indonesian when shopping online. The survey was conducted to 763 respondents and
produced result that 74.97% of the respondents use bank transfer, 13.37% use Cash
On Delivery, 7.73% use credit card, 1.57% use e-money, 0.79% use minimarket
payment, and 1.57% choose other payment method.
a) Please make an interval estimation for population proportion of Indonesian online
shoppers who use bank transfer, with 95% confidence level. Please perform
assumptions check before you use the appropriate method.
b) According to the data, is there sufficient evidence to state that in 2015, less than
80% of Indonesian online shoppers use bank transfer as payment method when
shopping online? (not included in midterm)
c) You are asked to update the study in 2017. Suppose that after two years, it is
known that only 70% of Indonesian online shoppers use bank transfer as payment
method. How many respondents are necessary to estimate the population
proportions within + 0.03, with 95% confidence level?

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 Solution
a. Assumption: amount of sample p and 1-p has
to be more than 5.
sample p = 0.7497 * 763 = 572.0211 = 573
sample 1-p = 763 – 573 = 190
Therefore, we can use sample proportion
estimation.
b. Since the the interval of proportion
estimation lies between 0.7189 and 0.7805,
then there is sufficient evidence that that in
2015, less than 80% of Indonesian online
shoppers use bank transfer as payment method
when shopping online.
 c.

Therefore, I need 457 respondents necessary to estimate the population

proportions
Good luck for your test! Do your best!

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