qt - assignment answer

MCP 1607 Reg. No: 213348645

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Contents

1. Introduction 2

2. Answer to Question 1 3









11. Marking Scheme and Comments 19

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1. Introduction

Quantitative Analysis is a vital part of any problem solving, be it from Engineering, Economics,

Logistics or Business Analysis. It enables us to glean information from statistical and other

analytical methods by which we can deduce much information that are otherwise not easily seen.

In business analysis, it is vital to extract information about correlation of data, i.e, whether the

sales quantities are related to weather, marketing poll results, changes in the economic landscape

or even changes in the political arena. To this end, it is necessary to perform regression analysis

before we can proceed any further. This assignment is focused on this area of data analysis/

The following sections comprise the answers to this assignment. This is a part of assignments for

Cohort No. 10.

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The following is the data table that is referred to in the ensuing answers.

Machine Age Fuel Consumption

Number (Years) (Ltr) 1 2 22 2 7.4 61 3 5.3 42 4 18 222 5 11.5 110 6 6.4 51 7 14.3 153 8 10.2 93 9 21 288

10 3.4 30 11 2.9 26 12 9.3 81 13 13.7 143 14 16.2 187 15 4.6 37

1. Plot the data on graph paper, consider age of machine as “X” and fuel consumption as “Y”

The plot is shown in the next page.

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2. Calculate the correlation coefficient between “x” and “y”.

r = …………… (1)

We refer the following table for this calculation:

Machine Age-x Fuel Consumption-y

Number (Years) (Ltr) xy x y x2 y2 1 2 22 44 2 22 4 484 2 7.4 61 451.4 7.4 61 54.76 3721 3 5.3 42 222.6 5.3 42 28.09 1764 4 18 222 3996 18 222 324 49284 5 11.5 110 1265 11.5 110 132.25 12100 6 6.4 51 326.4 6.4 51 40.96 2601 7 14.3 153 2187.9 14.3 153 204.49 23409 8 10.2 93 948.6 10.2 93 104.04 8649 9 21 288 6048 21 288 441 82944 10 3.4 30 102 3.4 30 11.56 900 11 2.9 26 75.4 2.9 26 8.41 676 12 9.3 81 753.3 9.3 81 86.49 6561 13 13.7 143 1959.1 13.7 143 187.69 20449 14 16.2 187 3029.4 16.2 187 262.44 34969 15 4.6 37 170.2 4.6 37 21.16 1369

∑ xy ∑ x ∑ y ∑ x2

∑ y2

21579.3 146.2 1546 1911.34 249880

])(].[)([

.

2222 yynxxn

yxxyn

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Where n = 15 Therefore, the correlation coefficient, r is calculated from eq. (1) to be r= (15*21579.3-146.2*1546)/SQRT((15*1911.34-146.2*146.2)*(15*249880-1546*1546)) r = 0.981160898 r ≈ 0.9812 (rounded to 4 decimal places)

3. Construct the regression equation of the from y = a + bx

We note that in the regression equation of the form

푦 = 푎 + 푏. 푥 ……………..(2)

used to derive the line (in a linear sense) of best fit, the coefficients are derived as

푏 = .(∑ . ) ∑ .∑.∑ (∑ ) ……………..(3)

푎 = 푦 − 푏. 푥̅ ……………..(4)

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Consider the following table values for this calculation:

Machine Age-x Fuel Cons.-y _ _

Number (Years) (Ltr) x y xy x y x2 y2 1 2 22 2 22 44 2 22 4 484 2 7.4 61 7.4 61 451.4 7.4 61 54.76 3721 3 5.3 42 5.3 42 222.6 5.3 42 28.09 1764 4 18 222 18 222 3996 18 222 324 49284 5 11.5 110 11.5 110 1265 11.5 110 132.25 12100 6 6.4 51 6.4 51 326.4 6.4 51 40.96 2601 7 14.3 153 14.3 153 2187.9 14.3 153 204.49 23409 8 10.2 93 10.2 93 948.6 10.2 93 104.04 8649 9 21 288 21 288 6048 21 288 441 82944 10 3.4 30 3.4 30 102 3.4 30 11.56 900 11 2.9 26 2.9 26 75.4 2.9 26 8.41 676 12 9.3 81 9.3 81 753.3 9.3 81 86.49 6561 13 13.7 143 13.7 143 1959.1 13.7 143 187.69 20449 14 16.2 187 16.2 187 3029.4 16.2 187 262.44 34969 15 4.6 37 4.6 37 170.2 4.6 37 21.16 1369

_ _

x y ∑ xy ∑ x ∑ y ∑ x2

∑ y2

9.746667 103.0667 21579.3 146.2 1546 1911.34 249880

From eq. (3), b is calculated as (rounded off to 4 decimal points)

푏 = 13.3866

From eq. (4), a is calculated as (rounded off to 4 decimal points)

푎 = −27.4084

Therefore, the regression equation for the linear best-fit of the above data points is

풚 = −ퟐퟕ.ퟒퟎퟖퟒ+ ퟏퟑ.ퟑퟖퟔퟔ ∗ 풙

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4. Draw the regression line on the same graph

The plot for this answer is given in the next page:

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5. Calculate R2 – value that measures the goodness of fit.

Referring to the values in the following table:

Using the equation

푅 = 1 − ……………..(5)

Where

sse = Sum of Squared Errors

=

= 3379.211028

and

sst = Sum of Squared Totals

=

= 90538.933333

Therefore, from eq. (5), the coefficient of determination is (rounded to 4 decimal places)

푅 = 0.9627

(푦 − 푦 ̂ )

(푦 − 푦 ̂ )

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6. In respect of each of the observations on age calculate the residual.

The residual values are calculated as per the following table.

Machine Age-x Fuel Cons.-y Residuals

Number (Years) (Ltr) x y xy x2 y2

1 2 22 2 22 44 4 484 -

0.635092644 22.63509264 2 7.4 61 7.4 61 451.4 54.76 3721 71.6527086 -10.6527086

3 5.3 42 5.3 42 222.6 28.09 1764 43.54078589 -

1.540785892 4 18 222 18 222 3996 324 49284 213.5509851 8.449014894

5 11.5 110 11.5 110 1265 132.25 12100 126.537891 -

16.53789102

6 6.4 51 6.4 51 326.4 40.96 2601 58.26607874 -

7.266078737

7 14.3 153 14.3 153 2187.9 204.49 23409 164.0204546 -

11.02045463 8 10.2 93 10.2 93 948.6 104.04 8649 109.1352722 -16.1352722 9 21 288 21 288 6048 441 82944 253.7108747 34.28912532

10 3.4 30 3.4 30 102 11.56 900 18.10618916 11.89381084 11 2.9 26 2.9 26 75.4 8.41 676 11.41287423 14.58712577

12 9.3 81 9.3 81 753.3 86.49 6561 97.08730533 -

16.08730533

13 13.7 143 13.7 143 1959.1 187.69 20449 155.9884767 -

12.98847671

14 16.2 187 16.2 187 3029.4 262.44 34969 189.4550514 -

2.455051359 15 4.6 37 4.6 37 170.2 21.16 1369 34.17014499 2.82985501

(푦 − 푦 ̂ ) 푦 ̂

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7. a) Calculate the mean of the residuals.

Mean of residuals =

= 1.56319401867222E-14

≈ 0

b) Draw histogram of the residuals and comment on the results (use three class intervals)

With regard to the residual range:

Maximum = 34.2891253155987

Minimum = -16.5378910201407

Span = 50.8270163357394

So the BIN ranges are as follows:

Range-1 = -16.5378910201407 - 0.40444775843908

Range-2 = 0.40444775843908 - 17.3467865370189

Range-3 = 17.3467865370189 - 34.2891253155987

Bin (Residual Value) Frequency 0.404447758 9 17.34678654 4 34.28912532 2

More 0

∑(푦 − 푦 ̂ )15

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Remarks:

1. The residuals do not show a normal distribution ( This violates the 1st assumption of regression

analysis).

2. Therefore, the linear regression model we selected is incorrect (not suitable) to represent and

analyze this data set.

08. Draw another graph with “x” as machine number and “y” as residual and comment on the result.

The plot is given in the next page:

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Remarks:

The residuals seem to be randomly distributed about zero (0).

The residuals do not seem to follow any pattern and seem to be independent of the

Machine Numbers.

09. Calculate the correlation coefficient between “x” (value of age) and residual and comment on the result.

We refer the following table with calculations for this answer:

Machine Age-x Residual-y

Number (Years) xy x y x2 y2 1 2 22.63509 45.27 2 22.635 4 512.35 2 7.4 -10.6527 -78.83 7.4 -10.65 54.76 113.48 3 5.3 -1.54079 -8.166 5.3 -1.541 28.09 2.374 4 18 8.449015 152.08 18 8.449 324 71.386 5 11.5 -16.5379 -190.2 11.5 -16.54 132.25 273.5 6 6.4 -7.26608 -46.5 6.4 -7.266 40.96 52.796 7 14.3 -11.0205 -157.6 14.3 -11.02 204.49 121.45 8 10.2 -16.1353 -164.6 10.2 -16.14 104.04 260.35 9 21 34.28913 720.07 21 34.289 441 1175.7 10 3.4 11.89381 40.439 3.4 11.894 11.56 141.46 11 2.9 14.58713 42.303 2.9 14.587 8.41 212.78 12 9.3 -16.0873 -149.6 9.3 -16.09 86.49 258.8 13 13.7 -12.9885 -177.9 13.7 -12.99 187.69 168.7 14 16.2 -2.45505 -39.77 16.2 -2.455 262.44 6.0273 15 4.6 2.829855 13.017 4.6 2.8299 21.16 8.0081

∑ xy ∑ x ∑ y ∑ x2 ∑ y2

4.64E-12 146.2 2.34E-13 1911.34 3379.211

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Correlation Coefficient, r is calculated as Using the values calculated in the above table and the following equation:

r =

r = 1.83514E-15 ≈ 0

Note: Please refer next page for the plot of the residuals vs. machine age.

Remarks:

1. The Correlation coefficient of zero means that the residuals are not linearly correlated with

machine age. Therefore, the residuals and the independent variables are independent of each

other.

2. Also, they are randomly distributed about their mean, 0. This means that it obeys the 2nd

assumption of regression analysis.

3. Therefore, our model of regression is correct.

4. However, from the plot given in the next page, it is obvious that they are non-linearly

correlated

5. Therefore, our linear regression model is incorrect

6. We should apply a non-linear regression model to analyze this set of data

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10. Marking Scheme and Comments

Learning Outcome/ Question Maximum Weightage First Marker Second Marker

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9

Total Marks 100%

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1st Marker’s comments: _____________________________________________________________

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Moderator’s comments: _____________________________________________________________

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