Regression Analysis for Margin Prediction

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Question 1
a. Preliminary analysis of the regression model
1) Descriptive statistics
Descriptive statistics
Margin Number Nearest Office Space Enrolment Income Distance Quality
Mean 46.4175 Mean 2977.616667 Mean 2.305833333 Mean 50.59333333 Mean 16.21416667 Mean 39.2 Mean 11.2495 Mean 0.566666667
Standard Error 0.712709414 Standard Error 41.70926984 Standard Error 0.078512928 Standard Error 1.542685703 Standard Error 0.39983767 Standard Error 0.432664858 Standard Error 0.445172422 Standard Error 0.045425676
Median 46.9 Median 2971 Median 2.3 Median 49.65 Median 16 Median 39 Median 12.08 Median 1
Mode 54.4 Mode 3003 Mode 2.2 Mode 44.9 Mode 15.5 Mode 40 Mode 12.16 Mode 1
Standard Deviation 7.807340464 Standard Deviation 456.902159 Standard Deviation 0.860066036 Standard Deviation 16.89927517 Standard Deviation 4.380002222 Standard Deviation 4.739606054 Standard Deviation 4.87661955 Standard Deviation 0.497613352
Sample Variance 60.95456513 Sample Variance 208759.5829 Sample Variance 0.739713585 Sample Variance 285.5855014 Sample Variance 19.18441947 Sample Variance 22.46386555 Sample Variance 23.78141824 Sample Variance 0.247619048
Kurtosis -0.392893447 Kurtosis 0.236234585 Kurtosis -0.470697081 Kurtosis -0.318372074 Kurtosis -0.503259346 Kurtosis -0.510651311 Kurtosis -0.46710657 Kurtosis -1.958680173
Skewness -0.162069201 Skewness -0.261272802 Skewness 0.015427532 Skewness -0.012656933 Skewness 0.038526881 Skewness 0.008437444 Skewness -0.036439358 Skewness -0.272487103
Range 39.2 Range 2601 Range 4.1 Range 73.5 Range 20.5 Range 22 Range 22.72 Range 1
Minimum 27.3 Minimum 1613 Minimum 0.1 Minimum 14 Minimum 6 Minimum 28 Minimum 0.32 Minimum 0
Maximum 66.5 Maximum 4214 Maximum 4.2 Maximum 87.5 Maximum 26.5 Maximum 50 Maximum 23.04 Maximum 1
Sum 5570.1 Sum 357314 Sum 276.7 Sum 6071.2 Sum 1945.7 Sum 4704 Sum 1349.94 Sum 68
Count 120 Count 120 Count 120 Count 120 Count 120 Count 120 Count 120 Count 120
Largest(1) 66.5 Largest(1) 4214 Largest(1) 4.2 Largest(1) 87.5 Largest(1) 26.5 Largest(1) 50 Largest(1) 23.04 Largest(1) 1
Smallest(1) 27.3 Smallest(1) 1613 Smallest(1) 0.1 Smallest(1) 14 Smallest(1) 6 Smallest(1) 28 Smallest(1) 0.32 Smallest(1) 0
Confidence Level(95.0%) 1.411235823 Confidence Level(95.0%) 82.58852007 Confidence Level(95.0%) 0.155463439 Confidence Level(95.0%) 3.05467177 Confidence Level(95.0%) 0.791718521 Confidence Level(95.0%) 0.856719632 Confidence Level(95.0%) 0.881485858 Confidence Level(95.0%) 0.089947376
From the descriptive statistics the appropriate estimate of the Margin will be
obtained from the mean, this gives a value of 46.4175. This value can be affected
with presence of outliers hence leading to inaccurate estimation of the margin
value.
The skewness of the Income data is 0.0084, this indicates that the data is
symmetrically distributed.
The sum of the quality variable indicates the number of motels that have been
built or extensively renovated in the past 6 years.
2) Set of correlation coefficient
Correlation coeffi cient
Margin Number Nearest Office Space Enrolment Income Distance Quality
Margin 1
Number -0.487627077 1
Nearest 0.188492588 0.064537585 1
Offi ce Space 0.588401934 -0.11525533 0.189820646 1
Enrolment 0.128399133 -0.059278379 0.038279504 0.035072974 1
Income 0.208150784 0.071863738 -0.051413296 0.117281667 -0.107853908 1
Distance -0.13058048 0.120497632 0.179926405 0.019342162 0.08452925 -0.103475854 1
Quality 0.399745297 -0.15268135 0.096276576 0.269762533 0.093060199 0.022803355 -0.149064745 1
Testing the hypothesis
H0 ; ρ=0
Vs
H0 ; ρ ≠0
z=r∗√n
α =0.05
Using the indicated formula, we can derive a table of z statistics as shown below

Obtaining the Z statistics
Margin Number Nearest Office Space Enrolment Income Distance Quality
Margin 10.95445115
Number -5.341686996 10.95445115
Nearest 2.064832851 0.706973821 10.95445115
Offi ce Space 6.445620243 -1.262558883 2.079380997 10.95445115
Enrolment 1.406542032 -0.649362112 0.419330958 0.384205176 10.95445115
Income 2.280177596 0.78722781 -0.563204437 1.284756296 -1.18148037 10.95445115
Distance -1.430437485 1.319985424 1.970995018 0.211882774 0.925971539 -1.133521192 10.95445115
Quality 4.378990325 -1.672540386 1.054657044 2.95510049 1.019423402 0.249798235 -1.632922462 10.95445115
At 5% significance level
We reject the null hypothesis if the value of the z statistics is less than 0.05 and
fail to reject otherwise.
From the tables calculated above the following pairs of variables can be
concluded to have a significant linear relationship.
Margin and (Nearest, Office space, Enrolment, Income, Quality)
Number and (Nearest, Income, Distance)
Nearest and (Office space, Enrolment, distance, Quality)
Office space and (Enrolment, Income, Distance, Quality)
Enrolment and (Distance, Quality)
Income and quality
3) Scatter diagrams
Margin vs Number

The regression coefficient of margin and number of given by -0.4876, when this is
compared against the scatter diagram it can be visualised that the two variables
have a weak negative correlation.
Margin vs Nearest
The regression coefficient of margin and Nearest is obtained as 0.1885.
From the scatter plot below, it can be deduced that the variables have a weak
positive association a factor which is supported by the regression coefficient.
Margin vs Office space
The regression coefficient of margin and office space is obtained as 0.5884, this
supports the scatter plot display below to indicate that the variables have a strong