Docstoc

Engineering Statistic Coursework - 2 way ANOVA and Linear Regression Report

Document Sample
Engineering Statistic Coursework - 2 way ANOVA and Linear Regression Report Powered By Docstoc
					                     ENGINEERING STATISTIC
                            COURSEWORK
       TWO WAY ANOVA AND LINEAR REGRESSION
                            (NOISE LEVEL)
MUHAMMAD AIZAT

AZMATUN BINTI ABIDIN

NURFAMIEZA BINTI RAMLE

NUR WASIMAH BINTI ABD. WAHAB

NURUL ZAWANI KAMILAH BINTI MOHAMAD

SUHADA SYAZA BINTI SAFIEE




Check out other articles and reports @ docstoc . Follow me @ Twitter
SUMMARY




        In this study, two ways ANOVA and regression analysis has been done in order the
relationship between the factors and the noise level. The noise level can be affected by location (for
example; noise level in the city is higher than in the rural community) and time or period (for
example; noise level in the night would be lower than the noise level in the morning and the
afternoon). The noise level measurement has been done in 3 different roads by using sound level
meter in order to know the level of noise that been exposed to the receiver, in order to know
whether a mitigation method should be taken or not. From the two ways ANOVA and analysis test, it
is proven that both location and period which act as factor has the relationship with the noise level
as both tests reject H0 which states that there are no relationship between noise level and location
or period.




Check out other articles and reports @ docstoc . Follow me @ Twitter
INTRODUCTION




        Noise can be defined as an unwanted or undesired sound. Decibel is the standard unit for
measurement of sound. Usually 80 db is the level at which sound becomes physically painful. And
can be termed as noise. Humans, animals, plants and even inert objects like buildings and bridges
have been victims of the increasing noise pollution caused in the world. Be it human or machine-
created, noise disrupts the activity and balance of life. While traffic dons the cap of being the largest
noise maker throughout the world, there are many others that add to it, making our globe
susceptible to its effects. The effect of noise pollution is multi-faceted and inter-related.




        Sources of noise pollution include Traffic noise as the main source of noise pollution caused
in urban areas. With the ever-increasing number of vehicles on road, the sound caused by the cars
and exhaust system of autos, trucks, buses and motorcycles is the chief reason for noise pollution.
Effects from this pollution are too much of noise disturbs the rhythms of working, thereby affecting
the concentration required for doing a work. Noise of traffic or the loud speakers or different types
of horns divert the attention, thus causing harm in the working standard. This pollution too indirectly
affects the vegetation. Plants require cool & peaceful environment to grow. Noise pollution causes
poor quality of crops.




        Noise pollution can measured in decibels. When noise is at 45 decibels, no human being can
sleep, and at 120 decibels the ear is in pain and hearing begins to be damaged at 85 decibels. In
noise pollution, sound level meter was used in this experiment to measured sound pressure level as
to get the data or result and it are commonly used in noise pollution studies for the quantification of
almost any noise, but especially for industrial, environmental and aircraft noise.




        Location of this noise pollution were located at busy road in urban areas whereby ever-
increasing number of vehicle on road at certain time. Location that we take as our experiment is
Jalan Gereja, Jalan Hang Tuah and Jalan Klebang Besar. This is because all these location have a
potentially to pollution.




Check out other articles and reports @ docstoc . Follow me @ Twitter
        In this experiment, we use a two method which are Two-way ANOVA and linear regression
to analyze our result. The two-way analysis of variance is an extension to the one-way analysis of
variance. There are two independent variables (hence the name two-way). The two independent
variables in a two-way ANOVA are called factors. While for linear regression is a general method for
estimating/describing association between a continuous outcome variable (dependent) and one or
multiple predictors (factors) in one equation.




Check out other articles and reports @ docstoc . Follow me @ Twitter
RESULTS




                                                   Location



                        Jalan Gereja          Jalan Hang Tuah           Jalan Klebang Besar


  Period
                Trial     Trial 2 Trial 3 Trial 1 Trial 2 Trial 3 Trial 1 Trial 2 Trial 3
                1         (dBa)    (dBa)   (dBa)   (dBa)      (dBa)   (dBa)   (dBa)   (dBa)
                (dBa)



  Morning

  (7.30am-      92.8      88.0     87.3    90.3    88.5       91.3    93.3    92.1    91.4
  9.30am)

  Afternoon

  (12pm- 1pm)   93.3      85.4     94.4    94.4    88.3       89.6    91.4    91.8    89.3



  Evening

  (4.30pm-      89.1      93.6     90.3    94.3    91.3       92.4    95.9    93.6    91.4
  6.30pm)

  Night

  (8pm-10pm)    83.4      86.2     84.4    86.3    84.1       85.2    84.2    85.8    84.8




TWO WAY ANOVA ANALYSES




Check out other articles and reports @ docstoc . Follow me @ Twitter
Step 1:




H0: there are no differences between the means for period

H1: the mean for period differs




Step 2:




SOURCE               DF                 SS                  MS        F₀

PERIOD               3                  273.890             91.2967   12.37

LOCATION             2                  25.627              12.8133   1.74

INTERACTION          6                  35.973              5.9956    0.81

ERROR                24                 177.140             7.3808

TOTAL                35                 512.630




Step 3:




                                     0.05,3,24




                              3.01




Check out other articles and reports @ docstoc . Follow me @ Twitter
Step 4:

Reject H0




Step 5:

Period does affect the noise level reading.




Step 1:




H0: There are no differences between the means for location

H1: The mean for location differs




Step 2:




SOURCE               DF                   SS                  MS        F₀

PERIOD               3                    273.890             91.2967   12.37

LOCATION             2                    25.627              12.8133   1.74

INTERACTION          6                    35.973              5.9956    0.81

ERROR                24                   177.140             7.3808

TOTAL                35                   512.630




Step 3:



Check out other articles and reports @ docstoc . Follow me @ Twitter
                                    0.05, 2, 24




                         3.40




Step 4:




Fail to reject H0




Step 5:




Location does not affect the noise level reading.

Step 1:




H0: There are no interaction between period and location

H1: Period and location interact




Step 2:



Check out other articles and reports @ docstoc . Follow me @ Twitter
SOURCE               DF                 SS            MS        F₀

PERIOD               3                  273.890       91.2967   12.37

LOCATION             2                  25.627        12.8133   1.74

INTERACTION          6                  35.973        5.9956    0.81

ERROR                24                 177.140       7.3808

TOTAL                35                 512.630




Step 3:




                                        0.05, 6, 24




                             2.51




Step 4:

Fail to reject H0.




Step 5:

Period and location does not interact




Check out other articles and reports @ docstoc . Follow me @ Twitter
REGRESSION ANALYSIS

Noise Level VS Location




              X               Y               X²   Y²         XY

              (LOCATION)      (NOISE LEVEL)

              1               90.725          1    8231.03    90.725

              2               89.058          4    7931.33    178.116

              3               89.3164         9    7977.42    267.95

TOTAL         6               269.0994        14   24139.75   536.791




Sxx = ∑X² - (∑X)²

                  n

   = 14 – (6)²

          3

   =2




Syy= ∑Y² - (∑Y)²

                  n

   =24139.81 – (269.0994) ²

                      3

   =1.6176




Sxy=∑ XY – (∑X) (∑Y)


Check out other articles and reports @ docstoc . Follow me @ Twitter
                       n

    =536.791 – (6) (269.0994) ²

                      3

    = - 1.4078




^

ß1 = Sxy

       Sxx



    = - 1.4078

         2




    = - 0.7039




^


ß0 = ỹ - ß1X



    = 89.6998 – (- 0.7039) (2)




    = 91.1076



Check out other articles and reports @ docstoc . Follow me @ Twitter
r=      Sxy

     √ Sxx Sxy



 =      - 1.4078

  √(2) (1.6176)




= - 0.7827




R² = 0.6126




     = 61.26 %

Noise Level VS Period




                 X          Y               X²   Y²         XY

                 (PERIOD)   (NOISE LEVEL)

                 1          90.56           1    8201.11    90.56

                 2          90.88           4    8259.17    181.76

                 3          92.43           9    8543.30    277.29

                 4          84.93           16   7213.10    339.72

TOTAL            10         358.8           30   32216.68   889.33




Sxx = ∑X² - (∑X)²

Check out other articles and reports @ docstoc . Follow me @ Twitter
                   n

    = 30 – (10)²

               4

    =5




Syy= ∑Y² - (∑Y)²

                   n

    = 32216.6419 – (358.8) ²

                       4

    = 32.32




Sxy=∑ XY – (∑X) (∑Y)

                       n



    = 889.33 – (10) (358.8)

                   4

    = - 7.67


^

ß1 = Sxy

         Sxx




Check out other articles and reports @ docstoc . Follow me @ Twitter
     = - 7.67

         5

     = -1.534




^

ß0 = ỹ - ß1X



     = 89.7 – (- 3.835)




     = 93.535




r=       Sxy

     √ Sxx Sxy



    =   - 7.67

     √(5) (32.32)




    = - 0.6037




R² = 0.3640




Check out other articles and reports @ docstoc . Follow me @ Twitter
  = 36.4 %




                                Period
         93

         92

         91

         90

         89
 Noise




         88

         87

         86
Regression Graph
         85

Title: Period Graph
     84
              1.0   1.5   2.0      2.5     3.0   3.5   4.0
                                  Period




Check out other articles and reports @ docstoc . Follow me @ Twitter
Title: Location Graph

                            Location
         90.8

         90.6

         90.4

         90.2

         90.0
 Noise




         89.8

         89.6

         89.4

         89.2

         89.0

                1.0   1.5        2.0      2.5   3.0
                               Location




CORRELATION ANALYSIS

Testing The Coefficient of Correlation

Period Factor


Check out other articles and reports @ docstoc . Follow me @ Twitter
Step 1:

H₀: ρ = 0 there is no correlation between period and noise level

H¡: ρ ≠ 0 there is a correlation between period and noise level




Step 2:

                      X                 Y               X²         Y²        XY

                   (PERIOD)          (NOISE
                                    LEVEL)

              1                 90.56               1         8201.11    90.56

              2                 90.88               4         8259.17    181.76

              3                 92.43               9         8543.30    277.29

              4                 84.93               16        7213.10    339.72

TOTAL         10                358.8               30        32216.68   889.33




t = r √ n-2

      √1- r²

 = - 0.6034 √4-2

 √ (1- (-o.6034)²




= - 0.8533

   0.7974

 = - 1.0701




Check out other articles and reports @ docstoc . Follow me @ Twitter
Step 3:

                                           t,5/2, 2




          -4.303                       4.303




Step 4: Reject H₀ at 5%




Step 5: There is a correlation between period and noise level




Location Factor

Step 1:



Check out other articles and reports @ docstoc . Follow me @ Twitter
H₀: ρ = 0 there is no correlation between location and noise level

H¡: ρ ≠ 0 there is a correlation between location and noise level




Step 2:

                     X                  Y              X²            Y²    XY

              (LOCATION)             (NOISE
                                    LEVEL)

                     1                90.725            1       8231.03   90.725

                     2                89.058            4       7931.33   178.116

                     3               89.3164            9       7977.42   267.95

TOTAL                6              269.0994           14      24139.75   536.791




t = r √ n-2

      √1- r²

 = - 0.7827 √3-2

 √ (1- (- 0.7827)²

 = - 0.7827

   0.6224

 = - 1.2576




Step 3:


                                     t, 5/2, 1



Check out other articles and reports @ docstoc . Follow me @ Twitter
          -12.71                  12.71




Step 4: Reject H₀ at 5%




Step 5: There is a correlation between location and noise level




TEST FOR LINEARLITY

Testing The Slope of the Regression Line



Check out other articles and reports @ docstoc . Follow me @ Twitter
Period factor

Step 1:

H₀: ß¡ = 0 there is no linear relationship between period and noise level

H¡: ß¡ ≠ 0 there is a linear relationship between period and noise level




Step 2:

                    X                   Y              X²          Y²        XY

                (PERIOD)             (NOISE
                                     LEVEL)

                    1                 90.56             1       8201.11     90.56

                    2                 90.88             4       8259.17     181.76

                    3                 92.43             9       8543.30     277.29

                    4                 84.93            16       7213.10     339.72

TOTAL               10                358.8            30      32216.68     889.33




  ^

t = ß¡- ß¡
      Se²


Check out other articles and reports @ docstoc . Follow me @ Twitter
    Sxx



=   (-1.534) – 0

     10. 2771

          5

= - 1.0699


              ^


Se² = Syy – ß¡Sxy

              n-2

    = 32.32 – (-1.534)(-7.67)

                  4-2

    = 10.2771




Step 3:




                                    t= 5/2, 2

Check out other articles and reports @ docstoc . Follow me @ Twitter

                  - 4.303       4.303
Step 4: Reject H₀ at 5%




Step 5: There is a linear relationship between period and noise level




LOCATION FACTOR

Step 1:

H₀: ß¡ = 0 there is no linear relationship between location and noise level


Check out other articles and reports @ docstoc . Follow me @ Twitter
H¡: ß¡ ≠ 0 there is a linear relationship between location and noise level




Step 2:

                        X                Y             X²          Y²         XY

                  (LOCATION)         (NOISE
                                     LEVEL)

                        1             90.725            1       8231.03      90.725

                        2             89.058            4       7931.33      178.116

                        3            89.3164            9       7977.42      267.95

TOTAL                   6            269.0994          14      24139.75      536.791


    ^

t = ß¡- ß¡

        Se²

        Sxx

=   (-0.739) – 0

         0.6266

              2

= - 1.2576




                   ^


Se² = Syy – ß¡Sxy

                  n-2



Check out other articles and reports @ docstoc . Follow me @ Twitter
    = 1.6176 – (-0.7039)(-1.4078)

                   3-2

     = 0.6266




Step 3:




                                    t =5/2 , 1




          -12.71                12.71




Step 4:

Reject H₀ at 5%




Step 5:

There is a linear relationship between location and noise level




CONCLUSION




Check out other articles and reports @ docstoc . Follow me @ Twitter
       The aim of noise level monitoring is to determine the exposure of noise to the receiver
(people). Several factors can affect the level of noise such as period and location of the
exposure. From the experiment that has been done, two factors have been tested for the
relationship with noise level. Theoretically, the noise level has relationship with location of
the exposure. If the location is the main road which more vehicles use the road, the noise
level would be high. For the period, the noise level would be higher during peak hour as more
people would use the road for work. The recorded data for the noise level monitoring has
been used for ANOVA two way analyses and correlation analysis. It is known that period and
location has significant relationship with noise level as both test reject H₀ at α = 5%.

       Two-way analysis of variance (ANOVA) measures the effect of two factors
simultaneously. On the hand, two ways ANOVA would not only be able to access both factors
in the same test but also whether there is an interaction between the factors. Meanwhile linear
regression analyzes the relationship between two variables, X and Y. The value r2 is a
fraction between 0.0 and 1.0, and has no units. An r2 value of 0.0 means that knowing X does
not help to predict Y. There is no linear relationship between X and Y, and the best-fit line is a
horizontal line going through the mean of all Y values. When r2 equals 1.0, all points lie
exactly on a straight line with no scatter.




RECOMMENDATION

In order to get an accurately and precisely data this experiment it should be enrolling at with
more various collection number of data.




APPENDIX




Check out other articles and reports @ docstoc . Follow me @ Twitter
        Picture 1.1: Sound Level Meter is an output port on the meter records sound

                                level data in decibels (dB).




Procedure of Data Analyzed using Two Way ANOVA
1. Step 1


Check out other articles and reports @ docstoc . Follow me @ Twitter
Click Minitab icon at desktop and page will be display as below:




Check out other articles and reports @ docstoc . Follow me @ Twitter
2. Step 2
Insert the experiment data as below:




Check out other articles and reports @ docstoc . Follow me @ Twitter
3. Step 3
Analyzed the data via MINITAB (Two Way ANOVA) as below:

   i)




Check out other articles and reports @ docstoc . Follow me @ Twitter
ii)




Check out other articles and reports @ docstoc . Follow me @ Twitter
4. Step 4
The analyzed data of Two Way ANOVA will had shown as below:




MINI TAB PROJECT

Two-way ANOVA: noise level versus period, location



Source         DF        SS        MS       F       P

period          3   273.890   91.2967   12.37   0.000

location        2    25.627   12.8133    1.74   0.198

Interaction     6    35.973    5.9956    0.81   0.571

Error          24   177.140    7.3808

Total          35   512.630



S = 2.717     R-Sq = 65.44%     R-Sq(adj) = 49.61%




5. Step 5

Proceed with Hypothesis Testing.




Check out other articles and reports @ docstoc . Follow me @ Twitter
Data Analyzed using Linear Regression
1. Step 1
Click Minitab icon at desktop and page will be display as below:




Check out other articles and reports @ docstoc . Follow me @ Twitter
2. Step 2
Insert the experiment data as below:




Check out other articles and reports @ docstoc . Follow me @ Twitter
3. Step 3
Analyzed the data via MINITAB (Linear Regression) as below:

   i)




Check out other articles and reports @ docstoc . Follow me @ Twitter
   ii)




Check out other articles and reports @ docstoc . Follow me @ Twitter
4. Step 4
The analyzed data of Linear Regression will had shown as below:




Regression Analysis: Noise versus Period

The regression equation is
Noise = 93.5 - 1.53 Period



Check out other articles and reports @ docstoc . Follow me @ Twitter
Predictor     Coef     SE Coef        T        P
Constant    93.535       3.928    23.81    0.002
Period      -1.534       1.434    -1.07    0.397


S = 3.20718    R-Sq = 36.4%        R-Sq(adj) = 4.6%


Analysis of Variance

Source            DF       SS       MS       F       P
Regression         1    11.77    11.77    1.14   0.397
Residual Error     2    20.57    10.29
Total              3    32.34



5. Step 5

Repeat Step 1 until Step 4 for Location data.

6. Step 6

Proceed with Hypothesis Testing.




DATA ON THE MINITAB

ANOVA-TWO WAY ANOVA

                         PERIOD             LOCATION     NOISE LEVEL
                        MORNING                A             92.8
                        MORNING                A             88.0
                        MORNING                A             87.3


Check out other articles and reports @ docstoc . Follow me @ Twitter
                  MORNING             B             90.3
                  MORNING             B             88.5
                  MORNING             B             91.3
                  MORNING             C             93.3
                  MORNING             C             92.1
                  MORNING             C             91.4
                 AFTERNOON            A             93.3
                 AFTERNOON            A             85.4
                 AFTERNOON            A             94.4
                 AFTERNOON            B             94.4
                 AFTERNOON            B             88.3
                 AFTERNOON            B             89.6
                 AFTERNOON            C             91.4
                 AFTERNOON            C             91.8
                 AFTERNOON            C             89.3
                  EVENING             A             89.1
                  EVENING             A             93.6
                  EVENING             A             90.3
                  EVENING             B             94.3
                  EVENING             B             91.3
                  EVENING             B             92.4
                  EVENING             C             95.9
                  EVENING             C             93.6
                  EVENING             C             91.4
                   NIGHT              A             83.4
                   NIGHT              A             86.2
                   NIGHT              A             84.4
                   NIGHT              B             86.3
                   NIGHT              B             84.1
                   NIGHT              B             85.2
                   NIGHT              C             84.2
                   NIGHT              C             85.8
                   NIGHT              C             84.8




REGRESSION ANALYSIS : NOISE LEVEL VS PERIOD – USING MINITAB



                           PERIOD         NOISE LEVEL
                              1               90.56
                              2               90.88
                              3               92.43


Check out other articles and reports @ docstoc . Follow me @ Twitter
                                  4               84.93




REGRESSION ANALYSIS : NOISE LEVEL VS LOCATION – USING MINITAB



                         LOCATION          NOISE LEVEL
                            1                  90.725
                            2                  89.058
                            3                 89.3164




REFERENCE

   1) Meet Minitab 15 for Windows, January 2007




Check out other articles and reports @ docstoc . Follow me @ Twitter

				
DOCUMENT INFO
Description: Application of Two Ways ANOVA and Linear Regression in real life