HI6007 Statistics for Business Decisions

A	B	C
6	5	6
5	5	7
4	4	6
5	4	5
6	5	6
4	4	6
5	5	6
4	6	6
6	5	4
5	6	5

1. State the null and alternative hypothesis for single factor ANOVA to test for any significant difference in the perception among three groups. (1 marks)
2. State the decision rule at 5% significance level.                                                           (2 marks)
3. Calculate the test statistic.                                                                                                      (6 marks) 
4. 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 rate	3.5	3.0	2.8	3.6	2.75	3.4	3.12	2.86	3.02	2.6	3.3
Housing starts	100	120	150	130	170	135	130	185	127	190	96

You are required to;

1. Derive the regression equation (3 marks)
2. 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	0.442
R Square	A
Adjusted R Square	0.080
Standard Error	2.587
Observations	25
ANOVA

	Df		SS	MS	F	Significance F
Regression		B	34.1036	E	F	0.1979
Residual		21	D	6.6933
Total		C	174.6631
	Coefficients		Standard Error	t Stat	Pvalue	Lower 95%

Intercept                                    12.31            4.70         2.62        0.02	2.54
Direct                                           0.57           1.72             H        0.74	-3.01
Newspaper                                   3.32           1.54         2.16        0.04	0.12
Television                                        G            1.96         0.37        0.71	-3.34
a. Complete the missing entries from A to H in this output		(2 marks)
b. Assess the independent variables significance at 5% level		(3 marks)
c. Does the model is significant at 5% level?		(2 marks)

FORMULA SHEET

Location of the p^th percentile: 

𝐿𝑝= ^𝑝 (𝑛+1)

100

IQR = Q₃ – Q₁

Expected value of a discrete random variable

𝐸

Variance of a discrete random variable

𝑉𝑎𝑟

Z and t formulas:

𝑥−𝜇 ̅ 𝜇 𝑝𝑥̅𝜇 𝑍 =  𝑍 𝑍 𝑡  _𝑠

𝜎𝑝𝑞

𝑛

Confidence intervals

Mean:

𝑠

𝑥̅  

̂

𝑝̂  

𝑧_𝛼²_/2 𝑝 𝑞

𝑛 = 𝐵₂                      

Time Series Regression 

𝑏

𝑏₀ = 𝑌 − 𝑏₁𝑡

𝑇_𝑡 = 𝑏₀ + 𝑏₁𝑡

 ANOVA:

𝑆𝑆𝑇𝑅

 MSTR

                       𝑘                                             ₂

 SSTR = ∑ 𝑛_𝑗(𝑥_𝑗̅ − 𝑥̿)

                      𝑗=1

𝑘

 SSE = ∑(𝑛_𝑗 − 1)𝑠_𝑗²

𝑗=1

Simple Linear Regression:

  ̂𝑦 = 𝑏₀ + 𝑏₁𝑥

                    ∑(𝑥_𝑖 − 𝑥̅)(𝑦_𝑖 − 𝑦̅)

       𝑏₁ =   ∑(𝑥_𝑖 − 𝑥̅)₂ SSE

  MSE =  

𝑛_𝑇 − 𝑘

𝑘        𝑛𝑗

2  SST = ∑ ∑(𝑥_𝑖𝑗 − 𝑥̿)

𝑗=1 𝑖=1

 F = MSTR / MSE

            𝑏₀ = 𝑦̅ − 𝑏₁𝑥̅

    SST    =    SSR    +    SSE

SSE = ∑(𝑦_𝑖 − 𝑦̂_𝑖)²                                  SST = ∑(𝑦_𝑖 − 𝑦̅)²

SSR= ∑(𝑦̂_𝑖 − 𝑦̅)²

Coefficient of determination 

     R²= SSR/SST                                                             

Correlation coefficient

 𝑌

𝑟      or            𝑟

R2 = (𝑟𝑥𝑦 )2

𝑟_𝑥𝑦 = (sign of 𝑏₁) Coefficient of Determination

Testing for Significance 

 s  = MSE = SSE/(n  2)                      s =  

  ^𝑠𝑏 𝑠                           ̅                                              ^{𝑡 =} 𝑠𝑏_𝑏11    F = MSTR / MSE

 MSR = SSR/k-1                 MSE = SSE/n-k

Confidence Interval for β₁

𝑏1 ± 𝑡𝛼/2𝑠𝑏₁

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                      p p

      ^{2                                                  2})     𝑛     

   𝑅_𝑎        = 1 − (1 − 𝑅

𝑛

R² = SSR/SST

\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.

Academic Integrity

Holmes Institute is committed to ensuring and upholding Academic Integrity, as Academic Integrity is integral to maintaining academic quality and the reputation of Holmes’ graduates. Accordingly, all assessment tasks need to comply with academic integrity guidelines.  Table 1 identifies the six categories of Academic Integrity breaches.  If you have any questions about Academic Integrity issues related to your assessment tasks, please consult your lecturer or tutor for relevant referencing guidelines and support resources.  Many of these resources can also be found through the Study Skills link on Blackboard.  

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 violation of this policy can incur a range of penalties varying from a 50% penalty through suspension of enrolment.  The penalty would be dependent on the extent of academic misconduct and your history of academic misconduct issues.  

All assessments will be automatically submitted to SelfAssign to assess their originality.

Further Information: 

For further information and additional learning resources please refer to your Discussion Board for the unit.