Statistical Analysis for Factors Influencing Pouring of Coffee in a Cup

StatisticalAnalysis for Factors Influencing Pouring of Coffee in a Cup

Descriptionof analysis

Inthis section, we will conduct an experiment to investigate theeffects of the explanatory variables on the dependent variable. Inour study as major interest will be to investigate factors thataffect coffee in the cup. In our study we will conduct an experimenton factors such as the amount of sugar in a cup regarding grams,walking speed of individual in meters per ssecond, type of cup inconsideration to its width and level of the coffee in milliliters. Inthis experiment we shall consider a sample size of 20 measurementstaken from each observation and then recorded in the table,(Montgomery, Runger, &amp Hubele, 2009).

Allour measurements in this data collection are continuous that willenable us to perform normality test in order to give us the way tocarry out any parametric test. In the analysis, descriptivestatistics will be used to give a summary to all variables that wereinvestigated in the experiment. This will be displayed in tabulatedforms with full details of each finding, (Merletti, &amp Parker,2004)

Otherpredictive analysis such ANOVA will be applied to compare meanswithin and between the treatments and check any significance betweenthem. This will be further elaborated on the F-statistics value andthe p-value. Inferential statistics such correlation will be used totest the strength of association between the dependent variable andexplanatory variables,( Tidd, Pavitt, &amp Bessant, 2001)

Regressionanalysis will also help to determine the effect of the independentvariables on the dependent variable and the effect size of thecoefficient parameters. Graphical data presentation will be anotherimportant aspect of analysis that explains more about the data in avisual manner. Some of the graphical methods that will be applied inthe analysis will include box plot, scatter plot, histogram, residualplot and QQ plot for test of normality, (Zhang., Wang, Ye2008)

Table1.0 Descriptive statistics for Amount of coffee in cup (milliliters)

Descriptive Statistics

&nbsp

&nbsp

&nbsp

Column 5 AMOUNT OF COFFE

Minimum: 50

&nbsp

&nbsp

1st Quartile:92.5

&nbsp

Sample Size, n: 20

2nd Quartile:160

&nbsp

Mean: 226.5

3rd Quartile:355

&nbsp

Median: 160

Maximum: 500

&nbsp

Midrange: 275

&nbsp

RMS: 273.8613

Sum: 4530

&nbsp

Variance, s^2: 24945

Sum Sq:1.500000e+6

St. Dev., s: 157.9399

Range: 450

&nbsp

Mean Abs Dev:140.15

Coeff. Of Var. 69.73%

95% CI for the Mean:

&nbsp

152.5818 &lt mean &lt300.4182

95% CI for the variance:

&nbsp

14426.8462 &lt VAR &lt 53214.5014

95% CI for the St. Dev.:

120.1118 &lt SD &lt 230.6827

FromTable 1.0 it clearly shows that the mean average of the amount ofcoffee that was an experiment in different cups of different sizeswas 226.5 milliliters with a confidence interval of 152.5818 and300.4182 as lower and upper limit intervals respectively with astandard deviation of 157 milliliters.

Thenormality plot is as shown below by histogram and box plot. The plotof the normality still is shown in the QQ plot that explains moreabout the linear association between the observed values.

Table1.1 Descriptive statistics for type of cup in width (cm)

Descriptive Statistics

Minimum: 35

&nbsp

Column 4 Type of cup(width)

1st Quartile:59.5

&nbsp

&nbsp

2nd Quartile:84

&nbsp

Sample Size, n: 20

3rd Quartile:175

&nbsp

Mean: 112.9

Maximum: 250

&nbsp

Median: 84

&nbsp

Midrange: 142.5

Sum: 2258

&nbsp

RMS: 132.0901

Sum Sq:348956

&nbsp

Variance, s^2: 4948.832

&nbsp

St. Dev., s: 70.34793

Range: 215

&nbsp

Mean Abs Dev:60.97

Coeff. Of Var. 62.31%

5% CI for the Mean:

95% CI for the variance:

152.5818 &lt mean &lt300.4182

14426.8462 &lt VAR &lt 53214.5014

95% CI for the St. Dev.:

120.1118 &lt SD &lt 230.6827

Thetable clearly indicates that the measurement of the speed that one ismoving while having the cup of coffee had a mean score of 7.8 metersper second with a standard deviation of 3.4.The mean age had aconfidence 95% confidence interval intervals of 6.1959 and 9.4041as lower and upper respectively.

Thenormality plot is clearly shown below by the boxplot, histogram andQQ plot. The QQ plot explains more about the linear trend between theobserved values.

Figure1.0 showing normality plot of the amount of sugar in the cup

Figure1.1 showing normality plot of the type of cup in terms of width

Table1.3 Descriptive statistics for walking speed (meters/sec)

Explore Data – Walking speed(meters/sec)

&nbsp

&nbsp

&nbsp

&nbsp

&nbsp

Sample Size, n: 20

Sum: 156

&nbsp

Mean: 7.8

&nbsp

Median: 7.5

95% CI for the Mean:

Midrange: 8

6.1959 &lt mean &lt9.4041

RMS: 8.485281

&nbsp

Variance, s^2:11.74737

95% CI for the St. Dev.:

St. Dev., s: 3.427443

2.6065 &lt SD &lt 5.006

&nbsp

Mean Abs Dev: 2.8

&nbsp

Range: 12

95% CI for the variance:

Coeff. Of Var.43.94%

6.794 &lt VAR &lt 25.0603

Maximum: 14

1st Quartile: 6

&nbsp

Minimum: 2

2nd Quartile: 7.5

&nbsp

3rd Quartile: 10

&nbsp

&nbsp

&nbsp

&nbsp

Thetable clearly indicates that the measurement of the speed that one ismoving while having the cup of coffee had a mean score of 7.8 metersper second with a standard deviation of 3.4.The mean rage had aconfidence95% confidence interval intervals of 6.1959 and 9.4041 as lower and upper respectively.

Thenormality plot is clearly shown below by the box plot, histogram andQQ plot. The QQ plot explains more about the linear trend between theobserved values.

Table1.4 Descriptive statistics amount of sugar in cup

Explore Data – Amount of sugar

Sum: 308

&nbsp

&nbsp

Sample Size, n: 20

95% CI for the Mean:

Mean: 15.4

12.5658 &lt mean &lt18.2342

Median: 15.5

&nbsp

Midrange: 15

95% CI for the St. Dev.:

RMS: 16.49242

4.6054 &lt SD &lt 8.8451

Variance, s^2:36.67368

&nbsp

St. Dev., s: 6.05588

95% CI for the variance:

Mean Abs Dev: 5

21.2101 &lt VAR &lt 78.235

Range: 20

2nd Quartile: 15.5

Coeff. Of Var.39.32%

3rd Quartile: 20

Minimum: 5

Maximum: 25

1st Quartile: 10

&nbsp

&nbsp

&nbsp

Table1.4 indicates the amount of sugar that was in cups. The mean score ofthe sugar amount was 15.4 grams and standard deviation of 6 grams.The mean score had 95% mean confidence interval of 12.5658and 18.2342 as lower and upper limits. The test of normality isindicated in the figure 1.4 from the histogram and QQ plot.

Figure1.3 showing the normality plot of walking speed of individual withcup of coffee

Figure1.4 showing normality tests for amount of sugar in the cup

TheQQ plot shows the linear trend of association between the variables. The line shows that there is linear association between the observedvalues of sugar in cup.

Table1.5 Descriptive statistics of level of coffee in cup in milliliters

Explore Data – Level of cup of coffee

Sum: 2465

&nbsp

&nbsp

Sample Size, n: 20

95% CI for the Mean:

Mean: 123.25

95.794 &lt mean &lt150.706

Median: 116

&nbsp

Midrange: 137.5

95% CI for the St. Dev.:

RMS: 135.8678

44.6141 &lt SD &lt 85.6843

Variance, s^2:3441.566

&nbsp

St. Dev., s: 58.66486

95% CI for the variance:

Mean Abs Dev: 42.9

1990.4165 &lt VAR &lt 7341.8003

Range: 225

Minimum: 25

Coeff. Of Var.47.60%

1st Quartile: 85

3rd Quartile: 142.5

2nd Quartile: 116

&nbsp

&nbsp

&nbsp

Maximum: 250

Table1.5 clearly indicates that the mean score of the level of coffee indifferent cups of different size was 123.25 with a standard deviationof 58.7 and 95% confidence of 95.794and 150.706 lower and upper limits respectively. The test ofnormality is depicted in the figures below. The QQ plot tries toexplain linear association between the observed values.

Figure1.5 showing normality tests between the observed values.

Table1.6 showing ANOVA table

Source: DF: SS: MS: Test Stat, F: Critical F: P-Value:

Treatment: 3 228387.3375 76129.112536.0852 2.724947 0.000

Error: 76160337.55 2109.704605

&nbsp

Total: 79388724.8875

&nbsp

&nbsp

Table1.6 confirms that the interaction of means within the dependentvariable and independent variable is significance with a p-value of&lt0.0001*.This is from Fisher information on the table that thevalue of F-statistics 2.724947 and critical value of 36.0852.Analysisof variance gives comparisons using a different treatment. This tellsus all our variables that are used to explain the pouring of coffeeare very strong explanatory variables and have very closerelationship.

RegressionModelling

Linearregression of Amount of coffee and individual factor.

Amountof coffee and Type of cup (width)

Theamount of coffee is defined by the type of cup (width) with a linearregression equation

Y= 0.0021X1+213.8, which implies that the intercept is 213.8 and the amount ofcoffee is predicted by the type of cup (width) by 0.0021 units forevery unit. The coefficient of determination R2=0.000007137 indicating that 0.00% of the data fits the model whichclearly indicates that the model is a bad model thus no relationshipbetween the amount of coffee and type of cup (width) and therefore,it cannot be used to predict the amount of coffee alone.

Amountof coffee and Walking speed

Theamount of coffee is defined by the walking speed with a linearregression equation

Y= 0.5618X2+219.06, which implies that the intercept is 219.06 and the amount ofcoffee is predicted by the walking speed by 0.5618 units for everyunit. The coefficient of determination R2=0.3153 equivalent to 31.53% which explains that 31.53% of the datafits the model, hence the model is a bad model for this data andtherefore, it cannot be used to predict the amount of coffee alone.

Amountof coffee and Amount of sugar

Theamount of coffee is defined by the amount of sugar with a linearregression equation

Y= 0.6236X3+222.23, which implies that the intercept is 222.23 and the amount ofcoffee is predicted by the amount of sugar by 0.6236 units for everyunit. The coefficient of determination R2=0.0094 which is equivalent to 0.94% and it explains that only 0.94%of the data fits the model, hence the model is a bad model for thisdata and therefore, it cannot be used to predict the amount of coffeealone.

Amountof coffee and Level of coffee

Theamount of coffee is defined by the level of coffee with a linearregression equation

Y= 0.961X4+97.494, which implies that the intercept is 97.494 and the amount ofcoffee is predicted by the level of coffee by 0.961 units for everyunit. The coefficient of determination R2=0.0453 and it explains that only 4.53% of the data fits the givenlinear regression model. This implies that this is a bad model forthe data and therefore, it cannot be used to predict the amount ofcoffee alone.

Table1.7 Multiple Regression Modelling

Regression Statistics

Multiple R

0.64236

R Square

0.412626

Adjusted R Square

0.244805

Standard Error

11.92799

Observations

19

&nbsp

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

361.0139

167.35

2.1572

0.0489

2.0784

719.949

2.07843

719.95

Type of cup(width)

-0.17166

0.1959

-0.8764

0.3956

-0.5917

0.24843

-0.5917

0.2484

Walking speed

-0.66925

0.2608

-2.5663

0.0224

-1.2286

-0.1099

-1.2286

-0.11

Amount of sugar

-0.95972

1.68

-0.5713

0.5769

-4.563

2.64359

-4.563

2.6436

Level of coffee

-0.88043

1.3417

-0.6562

0.5223

-3.7581

1.99721

-3.7581

1.9972

Number of columns used:

&nbsp

&nbsp

&nbsp

Dependent column: coffee in cup

Total Variation: 473955

&nbsp

Explained Variation: 112919

Intercept b0: 390.2004

Walking speed b1: -3.158828

Standard Error:155.1421

Amount of sugar , b2: -11.33396

Coeff of Det, R^2:0.2382484

Level of the coffee b3: 0.3327429

Adjusted R^2: 0.0351146

Width of the cup, b4: -0.0489733

P Value: 0.003620063

Theamount of coffee spilled (y) is affected by all the factors presentwhich are the walking speed b1, amountof sugar b2, level of the coffee b3, and finally width of the cupb4. Nowthe regression equation that will define how the amount of coffeespilled (y) is affected by each factor will be written in the form:

Y= 361.0139 – 0.17166b1 – 0.66925b2 -0.95972b3 – 0.88043b4.

Thisis known as the best line that fits the regression model, butremember that the coefficient of determinant is only 0.412626 whichis equivalent to 41.26%, implying that only 41.26% of the data wehave fits the regression model we have and therefore, the model isnot a good model.

Inorder to be able to set prediction about the dependent andindependent variables a multiple linear regression was fitted. Thisregression modeling was important in that given one variable one wasable to predict the effect size of it on dependent variables when allother factors are kept constant. From the regression model, it isclearly our data can only explain 3.5% of the model as indicatedadjusted R2 and the other parts are explained by the intercept. Witha p-value of: 0.003620063 which is very significant.

Fromthe coefficient of the walking speed, we can see that coffee in thecup has a negative association with speed. This can be statisticallyinterpreted that unit increase in speed leads to the purring ofcoffee by 3.1 units. This association is significant in that from atheoretical point of view if speed the amount of coffee in cupdecreases.

Theamount of sugar is another explanatory variable that can be used totell more about the pouring of coffee in the cup. This also shows anegative association in that if the amount of sugar in cup increasesthere would be a decrease in the amount of coffee by 11 units andthis association is significant.

Thewidth of the cup that contains the coffee is another strongexplanatory variable in that it also shows a negative associationwith the amount of coffee. If the width is bigger, there is bigsurface area thus the pouring of the coffee is lower while if thewidth is the small pouring of coffee is by: -0.0489733.

Conclusionand recommendation

Fromthe experiment, we have been able to investigate some of theparameters that affect pouring of coffee in the cup. From the generallinear regression, we have found that amount of sugar, walking speed,the level of the coffee in the cup are very significant factors thatare explaining pouring of coffee. I recommend further research workto be done to investigate about other factors that contribute topouring of coffee as our model can only explain 3.5%.

References

Montgomery,D. C., Runger, G. C., &amp Hubele, N. F. (2009). Engineeringstatistics.John Wiley

&ampSons.

Merletti,R., &amp Parker, P. A. (2004). Electromyography:physiology, engineering, and non

invasiveapplications(Vol. 11). John Wiley &amp Sons.

Tidd,J., Pavitt, K., &amp Bessant, J. (2001). Managinginnovation(Vol. 3). Chichester: Wiley.

Murray-Smith,R., &amp Johansen, T. (Eds.). (1997). Multiplemodel approaches to nonlinear

modellingand control.CRC press.

Zhang,H., Wang, J., &amp Ye, S. (2008). Predictions of acidity, solublesolids and firmness of pear using electronic nose technique. Journalof Food Engineering,86(3),370-378.