Theoretical Probability Questionnaire

  

Description

Assessment Task – Tutorial Questions
Unit Code: HI6007
Unit Name: Statistics for Business Decisions
Assignment: Tutorial Questions Assignment
Due: week 13 (23rd of Feb 2021)
Weighting: 50%
Purpose: This assignment is designed to assess your level of knowledge of the key topics covered in
this unit
Unit Learning Outcomes Assessed.:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Understand appropriate business research methodologies and how to apply them
to support decision-making process.
Understand various qualitative and quantitative research methodologies and
techniques.
Explain how statistical techniques can solve business problems;
Identify and evaluate valid statistical techniques in a given scenario to solve
business problems;
Explain and justify the results of a statistical analysis in the context of critical
reasoning for a business problem solving
Apply statistical knowledge to summarize data graphically and statistically, either
manually or via a computer package;
Justify and interpret statistical/analytical scenarios that best fits business solution;
Explain and justify value and limitations of the statistical techniques to business
decision making and;
Explain how statistical techniques can be used in research and trade publication
Description: Each week students were provided with three tutorial questions of varying degrees of
difficulty. The tutorial questions are available in the Tutorial Folder, for each week, on Blackboard.
The interactive tutorials are designed to assist students with the process, skills and knowledge to
answer the provided tutorial questions. Your task is to answer a selection of tutorial question for
weeks 1 to 11 inclusive and submit these answers in a single document.
The questions to be answered are;
Question 1
(7 marks)
a. With your own words, using relevant examples briefly define types of probability assigning
methods
(3 marks)
b. Transport trade association conducted a survey of their members to determine what they felt
were the important issues to be discussed with the management. The survey results showed
that 74% felt that the job security was the important issue, while 65% felt that salary
increment was an important issue. Of those who felt salary increment was an important issue,
60% also felt that job security was an important issue.
i.
What percentage of the members felt that both job security and salary increment
were important?
(2 marks)
ii.
What percentage of the members felt that at least one of these two issues was
important?
(2 marks)
Question 2
(7 marks)
Annual food consumption survey shows that number of instant food meals consumed per month by
university students is normally distributed with a mean of 10 and a standard deviation of 3.
a. Calculate the proportion of students consume more than 12 instant meals per month?
b. Estimate the probability that in a random sample of 25 students’ more than 275
instant meals are consumed.
Question 3
(11 marks)
D Dax limited installed a new safety equipment in order to reduce the number of person hours lost as
a result of industrial accident. In a test to of the effectiveness of the equipment, a random sample of
50 departments was chosen. The number of person- hours lost in the month prior to and the month
after the installation of the safety equipment was recorded. The percentage change was calculated
and recorded. Assume that the population standard deviation is 5 and sample mean is -1.2. Can we
infer at the 10% significance level that the new safety equipment is effective?
You are required to
a.
b.
c.
d.
e.
Formulate hypotheses
Decide the suitable test statistics and justify your selection.
Calculate the value of the relevant test statistics and identify the P value
Based on the test statistics in part (III), decide the decision criteria.
Make the final conclusion based on the analysis.
(3 marks)
(1 mark)
(3 marks)
(2 marks)
(2 marks)
Question 4
(11 marks)
The table below shows data on research to examine the perception of business ethics among 3 groups
of employees (higher score indicates higher ethical values).
A
6
5
4
5
6
4
5
4
6
5
B
5
5
4
4
5
4
5
6
5
6
C
6
7
6
5
6
6
6
6
4
5
a. State the null and alternative hypothesis for single factor ANOVA to test for any significant
difference in the perception among three groups.
(1 marks)
b. State the decision rule at 5% significance level.
(2 marks)
c. Calculate the test statistic.
(6 marks)
d. Based on the calculated test statistics decide whether any significant difference in the mean
price of gasoline for three bands.
(2 marks)
Note: No excel ANOVA output allowed. Students need to show all the steps in calculations.
Question 5
(7 marks)
Relax mortgage has gathered following data to examine the relationship between housing starts and
mortgage interest rate.
Interest 3.5
rate
Housing 100
starts
3.0
2.8
3.6
2.75
3.4
3.12
2.86
3.02
2.6
3.3
120
150
130
170
135
130
185
127
190
96
You are required to;
i.
ii.
iii.
Derive the regression equation
(3 marks)
Estimate the no of housing starts if mortgage interest rate is 2.5% (2 marks)
Calculate and interpret the correlation between interest rate and no of
housing starts.
(2 marks)
Question 6
(7 marks)
D& T LTD marketing team needed more information about the effectiveness of their 3 main mode of
advertising. To determine which type is the most effective, the manager collected one week’s data
from 25 randomly selected stores. For each store, the following variables were recorded:
Weekly gross sales
Weekly expenditure on direct mailing (Direct)
Weekly expenditure on newspaper advertising (Newspaper)
Weekly expenditure on television commercials (Television)
Following is the regression output based on the above-mentioned data.
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.442
A
0.080
2.587
25
ANOVA
Df
Regression
Residual
Total
B
21
C
SS
34.1036
D
174.6631
Intercept
Direct
Newspaper
Television
Coefficients
12.31
0.57
3.32
G
Standard
Error
4.70
1.72
1.54
1.96
F
Significance
F
0.1979
Pvalue
0.02
0.74
0.04
0.71
Lower 95%
2.54
-3.01
0.12
-3.34
MS
F
E
6.6933
t Stat
2.62
H
2.16
0.37
a. Complete the missing entries from A to H in this output
b. Assess the independent variables significance at 5% level
c. Does the model is significant at 5% level?
(2 marks)
(3 marks)
(2 marks)
FORMULA SHEET
K = 1 + 3.3 log10 n
Summary Measures (n – sample size; N – Population size)
∑𝑁
𝑖=1 𝑋𝑖
𝜇=
𝑁
𝑠2 =
1
𝑛−1
𝜎2 =
1
𝑁
∑𝑛
𝑖=1 𝑋𝑖
𝑝̂ =
𝑛
∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )2
1
[(∑𝑛𝑖=1 𝑥𝑖2 ) −
𝑛−1
Or 𝑠 2 =
𝑠~
𝑋̅ =
𝑋
𝑛
Or 𝑠 2 =
(∑𝑛
𝑖=1 𝑥𝑖 )
𝑅𝑎𝑛𝑔𝑒
]
1
𝑁
2
2
[(∑𝑁
𝑖=1 𝑥𝑖 ) − 𝑛µ ]
𝐶𝑉 =
4
[(∑𝑛𝑖=1 𝑥𝑖2 ) − 𝑛𝑥̅ 2 ]
2
𝑛
2
2
∑𝑁
𝑖=1(𝑥𝑖 − µ) Or 𝜎 =
1
𝑛−1
𝜎
µ
𝑠
𝑐𝑣 = 𝑥̅
Location of the pth percentile:
𝐿𝑝=
𝑝
(𝑛+1)
100
IQR = Q3 – Q1
Expected value of a discrete random variable
𝐸(𝑥) = 𝜇 = ∑ 𝑥 ∗ 𝑓(𝑥)
Variance of a discrete random variable
𝑉𝑎𝑟(𝑥) = ∑(𝑥 − 𝜇)2 𝑓(𝑥)
Z and t formulas:
𝑍=
𝑥−𝜇
𝜎
𝑍=
𝑥̅ −𝜇
𝜎
√𝑛
Confidence intervals
Mean:
𝑥̅ ± 𝑧𝛼/2
𝜎
√𝑛
𝑍=
𝑝̂−𝑝
𝑝𝑞
√
𝑛
𝑡=
𝑥̅ −𝜇
𝑠
√𝑛
𝑥̅ ± 𝑡𝛼/2
𝑠
√𝑛
Proportion:
𝑝̂ 𝑞̂
𝑝̂ ± 𝑧𝛼 √
𝑛
2
𝑛=
2
𝑧𝛼/2
𝑝𝑞
𝐵2
Time Series Regression
𝑏1 =
∑𝑛𝑡=1[(𝑡 − 𝑡)(𝑦𝑡 − 𝑦)]
∑𝑛𝑡=1(𝑡 − 𝑡)
2
𝑏0 = 𝑌 − 𝑏1 𝑡
𝑇𝑡 = 𝑏0 + 𝑏1 𝑡
ANOVA:
𝑆𝑆𝑇𝑅
MSTR =
𝑘−1
MSE =
𝑘
SSTR = ∑ 𝑛𝑗 (𝑥̅𝑗 − 𝑥̿ )
SSE
𝑛𝑇 − 𝑘
𝑘
2
𝑗=1
𝑛𝑗
SST = ∑ ∑(𝑥𝑖𝑗 − 𝑥̿ )
2
𝑗=1 𝑖=1
𝑘
SSE = ∑(𝑛𝑗 − 1)𝑠𝑗 2
F = MSTR / MSE
𝑗=1
Simple Linear Regression:
̂𝑦 = 𝑏0 + 𝑏1 𝑥
𝑏1 =
∑(𝑥𝑖 − 𝑥̅ )(𝑦𝑖 − 𝑦̅)
∑(𝑥𝑖 − 𝑥̅ )2
SSE = ∑(𝑦𝑖 − 𝑦̂𝑖 )2
SSR= ∑(𝑦̂𝑖 − 𝑦̅)2
Coefficient of determination
𝑏0 = 𝑦̅ − 𝑏1 𝑥̅
SST
=
SSR
SST = ∑(𝑦𝑖 − 𝑦̅)2
+
SSE
R2= SSR/SST
Correlation coefficient
𝑟=
∑(𝑥− 𝑥̅ )(𝑦− 𝑦)
𝑟=
or
√(∑(𝑥− 𝑥)2 )(∑(𝑦− 𝑦)2 )
∑ 𝑋𝑌−
√(∑ 𝑋 2 −
∑𝑋 ∑𝑌
𝑁
(∑ 𝑋)2
(∑ 𝑌)2
)(∑ 𝑌 2 −
)
𝑁
𝑁
R2 = (𝑟𝑥𝑦 )2
𝑟𝑥𝑦 = (sign of 𝑏1 )√Coefficient of Determination
Testing for Significance
2
s = MSE = SSE/(n
𝑠𝑏1 =
SSE
s = √MSE = √𝑛−2
2)
𝑠
√∑(𝑥𝑖 −
𝑡=
𝑥̅ )2
MSR = SSR/k-1
MSE = SSE/n-k
Confidence Interval for β1
𝑏1 ± 𝑡𝛼/2 𝑠𝑏1
Multiple Regression:
y =  +  x +  x +. . . +  x + 
0
1 1
2 2
p p
𝑦̂ = b + b x + b x + . . . + b x
0
1 1
2 2
𝑅𝑎 2 = 1 − (1 − 𝑅 2 )
R2 = SSR/SST
𝑏1
𝑠𝑏1
p p
𝑛−1
𝑛−𝑝−1
F = MSTR / MSE
F distribution
Submission Directions:
The assignment will be submitted via Blackboard. Each student will be permitted only ONE submission
to Blackboard. You need to ensure that the document submitted is the correct one.
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Holmes Institute is committed to ensuring and upholding Academic Integrity, as Academic Integrity is
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categories of Academic Integrity breaches. If you have any questions about Academic Integrity issues
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Academic Integrity breaches are a serious offence punishable by penalties that may range from
deduction of marks, failure of the assessment task or unit involved, suspension of course enrolment,
or cancellation of course enrolment.
Table 1: Six categories of Academic Integrity breaches
Plagiarism
Reproducing the work of someone else without attribution. When
a student submits their own work on multiple occasions this is
known as self-plagiarism.
Collusion
Working with one or more other individuals to complete an
assignment, in a way that is not authorised.
Copying
Reproducing and submitting the work of another student, with or
without their knowledge. If a student fails to take reasonable
precautions to prevent their own original work from being copied,
this may also be considered an offence.
Impersonation
Falsely presenting oneself, or engaging someone else to present as
oneself, in an in-person examination.
Contract cheating
Contracting a third party to complete an assessment task,
generally in exchange for money or other manner of payment.
Data fabrication and
falsification
Manipulating or inventing data with the intent of supporting false
conclusions, including manipulating images.
Source: INQAAHE, 2020
If any words or ideas used the assignment submission do not represent your original words or ideas,
you must cite all relevant sources and make clear the extent to which such sources were used.
In addition, written assignments that are similar or identical to those of another student is also a
violation of the Holmes Institute’s Academic Conduct and Integrity policy. The consequence for a
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Further Information:
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the unit.

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