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Milk Price Investigation

MAB233 Engineering Mathematics 3
Group Report

Due Date: 5th November, 2012
Table of Contents Executive Summary 5 1.0 Introduction 6 2.0 Data Collection 6 2.1 Context and issues of interest 6 2.2 Planning and collection of data 6 2.2.1 Planning 6 2.2.2 Collection of variables 7 2.3 Quality of data and discussion 8 3.0 Data Exploration 9 3.1 Handling and processing data 9 3.2 Exploring and analysis 10 3.2.1 Histogram of unit price 10 3.2.2 Stem-and-leaf of price and unit price 11 3.2.3 Descriptive Statistics of price, unit price with type 12 3.2.4 Scatterplot of unit price, quality guarantee period and type of milk 13 3.2.5 Dotplot of price 14 3.2.6 Boxplot of unit price 15 3.2.7 Bar chart of container size and packaging 15 3.2.8 Pie chart of brand 16 3.3 Observation and commenting 16 4.0 Data analysis 17 4.1 One way ANOVA analysis 17 4.1.1 Price versus Packaging 17 4.1.2 Price versus Quality guarantee period 19 4.1.3 Unit price versus Type 21 4.1.4 Unit price versus Brand 23 4.2 Two way ANOVA analysis 26 4.2.1 Price versus Quality guarantee period and Type 26 4.2.2 Unit price versus Store and Brand 28 4.3 Regression 30 5.0 Conclusion 31 Reference 32 Appendix 33 Appendix A 33 Appendix B 34

List of Illustrations Figure 1. Planning for data collection 7 Figure 2. Grouped histogram of unit price 10 Figure 3. Stem-and-leaf of unit price 11
Figure 4. Stem-and-leaf of price…………………………………………………………………………………………………12 Figure 5. Descriptive Statistics of price and unit price 12 Figure 6. Scatterplot of unit price VS quality guarantee period 13 Figure 7. Dotplot of price 14 Figure 8. Boxplot of unit price by quality guarantee period 15 Figure 9. Bar chart of container size, packaging 16 Figure 10. Pie chart of brand 16 Figure 11. One-way ANOVA: Price versus Packaging 17 Figure 12. Residual plots for price 18 Figure 13. Boxplot of price 19 Figure 14. Price versus quality guarantee period 20 Figure 15. Residual plots of price 20 Figure 16. Boxplot of price 21 Figure 17. Unit price versus type 22 Figure 18. Residual plots for unit price 22 Figure 19. Boxplot of unit price 23 Figure 20. Unit price versus brand 24 Figure 21. Residual plots of unit price 24 Figure 22. Boxplot of unit price 25 Figure 23. Price versus type, quality guarantee 26 Figure 24. Residual plots of price 27 Figure 25. Unit price versus store, brand 28 Figure 26. Residual plots for unit price 29 Figure 27. Regression analysis of Quality guarantee period vs Unit price and Container size 30 Figure 28. Residual plots for quality guarantee period 31

Executive Summary
This project is mainly talk about the factor which can influence the price of milk by use the statistical data analysis from the mathematics lecture. The group decides to analysis the factor such as store, brand, milk type, container size, quality guarantee period and packaging. In order to analysis this data, our group use the three steps to achieve this aim. The first is identifying and describing a context and issues of interest, planning and collecting of relevant data and quality of data and discussion of context. The second is handled and process data, summarize exploring and comment on features of the data and statistical model. Finally, the group use statistical tools for statistical analysis and interpretation of the data in the context.

| Strong evidence | Some evidence | Insufficient evidence | Price vs packaging | √ | | | Price vs quality guarantee period | | | √ | Unit price vs type | √ | | | Unit price vs brand | √ | | | Price vs type and quality guarantee period | | | √ | Unit price vs store and brand | √ | | | Quality guarantee period vs Unit price & Container size | √ | | |

The table above proves that many variables can affect the price of milk, but some of them may not influence the price of milk such as the Quality guarantee period. There is no evidence can prove it can influence the price. If our group collects more data of milk can more clear to analysis.

1.0 Introduction
Milk is a white liquid and comes from the mammary glands of mammals, in 2011 the world’s dairy farms produced about 730 million tonnes of milk (Food outlook 2012). It has high nutrition, it also the primary source for young mammals when they cannot digest other types of food, Australia, New Zealand and United States are the world’s largest exporters of milk and milk products, so, milk price is a hot topic be more concern at present because of different factors decide the different milk price, like brand and packaging.

The main purpose of this report is analysis the relationship between milk prices with these factors using Minitab, like histogram, dotplot, scatterplot and boxplot, moreover, analysis through one-way ANOVA and two-way ANOVA to explore effects, further analysis the relationship with these variables and try to find which have strong evidence, some evidence and insufficient evidence.
There are six main factors which can affect the price of milk, they are store, brand, milk type, container size, quality guarantee period and packaging.
2.0 Data Collection
2.1 Context and issues of interest
Milk is a basic food in our life and the most of people drink milk in the morning. This project was based on milk price and unit price from each market and retail store. Everyday people drink the milk but not many people know the factor can influence the milk price. So our group was interested the effective factor which can influence the price of milk. According to discuss, there are six factor can influence the price and unit price of milk such as store, brand, milk type, container size, quality guarantee period and packaging. They are categorical variables, ordinal variables and continuous variable. Not only the individual factor can influence the price but also these factors interacted and affected the price and unit price. In order to know the way this factor can influence the price and unit price. The group members use the form to collect data and analysis it by Minitab software.
2.2 Planning and collection of data
2.2.1 Planning
At first, milk always can be distributed in many supermarkets and stores, however, well-know or bigger stores have more comprehensive milk like brand and types. So, visit these stores as our almost data, like Coles and Woolworths is the best choice.
Moreover, milk as the product being replaced because they always have shorter quality guarantee period than others, so finish it in half a day or several hours can keep the accuracy.

Next, take notes and take photo as the main way for record, because they have more advantages which are clear and easy to understand.

Therefore, combine them together to be a whole and prefect plan (Figure 1).

Figure 1. Planning for data collection
2.2.2 Collection of variables
There are eight main factors are shown below:

1. Store: Categorical variable
Store is a public area which can sell something like milk, different kinds of store decide the price, the group chooses four stores as samples, they are Convenience store, Coles, Seven-eleven and Woolworths.

2. Brand: Categorical variable
Brand is name for different types of milk, it is important element to recognize them, ten types of brand be record, they are A2, Coles, Dairy farmers, Organic, Pauls, Pure organic, Queensland, Seven-eleven, Vitasoy and Woolworths, so brand is categorical variable.

3. Type: Ordinal variable
When the milk is processed, it can be divide many types, there are sixteen types in database, they are fat free, full cream, low fat, new, fat free (more calcium), low fat (more calcium), soy regular, soy lite, rice, soy high fibre, strawberry, goat, kids omega3, low fat (+vitamins A&D), extra cream and shape.

4. Container size (L): Ordinal variable
It is a specification for milk, in normal stores milk always is divide 0.50L, 1.00L and 2.00L, so this is ordinal variable. Six kinds of container size are found, 2.00L, 1.00L, 0.60L, 0.30L, 3.00L and 0.50L.

5. Price ($): Continuous variable
Different price decide the quality of milk, it is also a comprehensive evaluation, the price is collected from 0.89 - 5.55 there are 118 products.

6. Quality guarantee period (days): Ordinal variable
This is a range about time for milk, in this range the milk will be very fresh, it is also an important factor to judge quality for milk. In these milk, the range is from two days – twelve days.

7. Unit price ($/L): Continuous variable
It can tell people the cost for per liter and it can be a good way of finding which the “best buy” is.
Based on this collection, from 1.000 – 4.790.

8. Packaging: Categorical variable
It is a good way for store milk. There are two types of packaging in this collection, paper and plastic.
According to these factors group members make a chart as the whole data for milk (Appendix 1).

2.3 Quality of data and discussion
Although the data for milk’s information were attentively collected in four different stores in one area as planned, in face fact some data may not influence acutely and also have some problem for the whole analysis. For example, the quality guarantee period is hard to find on the package of the milk as it is just imprinted the expired date on the cover but no date of production. In addition, every member cannot calculate and conjecture the quality guarantee period because milk was put on the counter every day. With this unexpected situation, the information of the average milk quality guarantee period was finally got from the staff of the supermarket. Furthermore, some type of milk such as goat milk, soy milk and special milk (more calcium or more vitamins) are not as general as the full cream, low fat and fat free milk. So the data for these kinds of milk are only five or less comparing with the huge data of the normal milk. Thus, it cannot be presented clearly in the further analysis. These problems will be fixed in next part by changing the details and value of the variables.
3.0 Data Exploration
The data exploration process utilised many graphical and numerical methods to gain understanding of the data set. All these exploration methods were conducted in MiniTab and observations made on the usefulness of the data to prove or disprove our hypotheses, the relations between variables and any shortcomings of data set.
3.1 Handling and processing data
The group’s members collect the data from market and retail store by take the photo. After the data has been put into excel, there are some variable is only have one or two elements and it may not influence the final result. But from the graph, there are many elements with only one or two number and it cannot make sense. There is an example for this problem. On the Dotplot of unit price the type of milk has the full cream, fat free, low fat, new, low fat with more calcium and low fat with Vitamins A&D and so on. But only few brands have the low fat with more calcium and low fat with Vitamins A&D. On this graph may not show any one of these types, but on the side of this graph it may shows that the red triangle is the low fat with calcium. So after group discuss, all the special type of milk are been change to others. After our group changes this, the others can show on the graph and the variable can more clear on other graph such as scatterplot and so on. In the same way group change some brand which only has few type of milks to other brands. Finally our group members rewrite the data into Minitab and doing the data analysis.
Here is the data sheet that has been changed (Appendix B).

3.2 Exploring and analysis
3.2.1 Histogram of unit price

Figure 2. Grouped histogram of unit price
From the histogram above (Figure 2), it shows that diverse distribution lines stand for different stores and it is obvious to see that the mean value of the green line reaches $2.927 per liter which means the unit price of the milk is highest in Seven-Eleven retail shop compared with other investigated stores. In comparison, Coles and Woolworths provide the lowest unit price of the milk; they are $2.01 per liter and $2.139 per liter respectively. Furthermore, the standard deviation illustrate that in Seven-Eleven retail shop the unit price of the milk has higher variation and it exist relative bigger dispersion from the mean value of the unit price than others. The other three stores’ standard deviation are all around 0.7 which is smaller than 0.9 from Seven-Eleven shop.

3.2.2 Stem-and-leaf of price and unit price

Stem-and-Leaf Display: unit price ($/L)

Stem-and-leaf of unit price ($/L) N = 118
Leaf Unit = 0.10

22 1 0000000011122444444444 53 1 6666666667777777778889999999999
(21) 2 000000001233333344444 44 2 5555555555666667777777777777789999 10 3 0012 6 3 99 4 4 1 3 4 677 1 5 5

Stem-and-Leaf Display: price ($)

Stem-and-leaf of price ($) N = 118
Leaf Unit = 0.10

2 0 88 7 1 02224 18 1 66799999999 40 2 0000000000001222333334
(30) 2 555556666777777777777789999999 48 3 00000012233334444444 28 3 55778899 20 4 444 17 4 677899999 8 5 0000001 1 5 5

Figure 3. Stem-and-leaf of unit price Figure 4. Stem-and-leaf of price
The stem-and-leaf plots which in figure N at the above present quantitative data for milk price in a graphical format. It can assist in visualizing the shape of a distribution, but it is different with histograms. The stemplot can show the original data such as milk price and put the data in order. It can show people the entire price in this plot and easy to analysis these data.

Descriptive Statistics: price ($), unit price ($/L)

Variable type N N* Mean SE Mean StDev Minimum Q1 price ($) fat free 15 0 2.853 0.233 0.902 1.260 1.960 full cream 40 0 2.821 0.187 1.183 0.890 2.000 low fat 27 0 3.188 0.215 1.118 1.750 2.250 new 27 0 3.110 0.199 1.035 1.630 2.230 others 5 0 3.388 0.583 1.305 2.040 2.345 soy 4 0 2.380 0.114 0.229 2.090 2.165

unit price ($/L) fat free 15 0 2.118 0.134 0.518 1.260 1.685 full cream 40 0 2.163 0.136 0.863 1.000 1.494 low fat 27 0 2.128 0.146 0.760 1.000 1.687 new 27 0 2.1465 0.0972 0.5050 1.1150 1.7150 others 5 0 3.388 0.583 1.305 2.040 2.345 soy 4 0 2.380 0.114 0.229 2.090 2.165

Variable type Median Q3 Maximum price ($) fat free 2.770 3.370 5.000 full cream 2.745 3.558 5.550 low fat 3.000 3.810 5.190 new 2.750 3.490 4.950 others 2.670 4.790 4.790 soy 2.390 2.585 2.650

unit price ($/L) fat free 1.950 2.750 2.870 full cream 1.995 2.660 4.667 low fat 2.000 2.700 3.900 new 2.0400 2.7000 2.9500 others 2.670 4.790 4.790 soy 2.390 2.585 2.650

3.2.3 Descriptive Statistics of price, unit price with type

Figure 5. Descriptive Statistics of price and unit price
The basic descriptive statistical analysis was displayed above in figure 5. To grasp the overall situation, it clearly emerges the mean value, median value and other important value of variables like the price and the unit price for different type of milk. Moreover, the ‘others’ type of milk is the most expensive one in the whole research which can be proved not only by the price but also by the unit price. If only focus on the mean value of the unit price, the ‘soy’ type of milk is $2.38 per liter ahead the other ones expect the ‘others’ one, following by the ‘full cream’, ‘new’, ‘low fat’ and ‘fat free’ type of milk.

3.2.4 Scatterplot of unit price, quality guarantee period and type of milk

Figure 6. Scatterplot of unit price VS quality guarantee period
This scatterplot is used to compare the quality guarantee period of different type of milk and at the same time determine the unit price level. Although the graph does not present clearly because too much reduplicate data lead to overlap happening, it still can explain what the graph account for. Most of the ‘full cream’ milk’s quality guarantee period with different unit price is 8 days. However, only one data is abnormal which means this milk will be overdue soon. In addition, the ‘new’ type of milk has many varieties, so the quality guarantee period concentrates on 7and 9 days, some of the data are marked in the column of 2, 8 and 12 days. Furthermore, the quality guarantee period of the ‘others’ type of milk and ‘soy’ milk are same for 7 days. But the unit price of the ‘others’ type of milk is much high than the ‘soy’ milk. Moreover, the ‘fat free’ milk’s quality guarantee period is the longest one which can reach 12 days and ‘low fat’ milk can be refrigerated for a long time (9 days) as well.

3.2.5 Dotplot of price

Figure 7. Dotplot of price
The dot plot which at above shows the plot of piece versus store with the different container size. This plot can clearly show the price from each store and plot the most prices in this store. The most popular of milk price is $2 and the most milk price is between $2 and $3. In these stores, 1L container size is the most popular size and large size may have the high price in each store.

3.2.6 Boxplot of unit price

Figure 8. Boxplot of unit price by quality guarantee period
The boxplot at the above shows the distribution of data. This plot can easy to compare the median and the quarter points. According to this plot, the others that include goat milk price are the highest price in these six types of milk. The soy milk price is more concentrate than other five types. Full cream milk price are very disperse. It has the lowest piece in all of the milk, but it also has the highest price in all of milk.
3.2.7 Bar chart of container size and packaging

Figure 9. Bar chart of container size, packaging
The figure 9 represents two categorical variables and they belong to packaging, they are plastic and paper. Obviously, the packaging of plastic as leader in large capacity milk, 0.50 liter is a special case, after 1.00 liter the packaging of paper has disappeared. In addition, 1.00 liter has both two types of packaging but plastic still have a little more advantages than paper. Moreover, in some small capacity milk, such as 0.30 and 0.60, paper as the main way for milk, there is no plastic packaging into these capacities.

3.2.8 Pie chart of brand

Figure 10. Pie chart of brand
This pie chart shows that the proportion of brand in Figure 10, they were very clear displayed. In these brands, Pauls has the highest market rate than others, it has almost 50%, moreover, Dairy farmers is the second large percent in it, it has almost 25%, Seven-eleven has the smallest proportion and it is similar with Organic, other brands which are A2, Coles, Pure organic, Queensland, Vitasoy and Woolworths, add them up are almost 25%.
3.3 Observation and commenting
Through the observation of the eight figures above, there are six deductions can be discovered. Firstly, the milk price in retail shops is higher than the supermarkets’. Therefore, Seven-Eleven retail shop and convenient shop’s milk price are ahead Woolworths and Coles. Secondly, the ‘other’ type of milk is more expensive than other ones which can both be illustrated in boxplot and descriptive statistics. Thirdly, the more expensive and more fat milk always have short quality guarantee period but do not eliminate the special price which is affected by the coming overdue day. Fourthly, the most popular price of milk is $2 and most of the price is concentrate between $2 and $3 in the whole milk market. In addition, the one liter container size of milk is accepted by most of the customers. Furthermore, plastic packaging almost occupy the total milk market, only some small size like 0.3L and 0.6L are paper packaging. However, the 1L container size of milk both has plastic and paper packaging. Finally, there are 10 brands of milk are investigated, but they do not get the same status in the milk market. Pauls milk gets nearly 50% and Dairy Fammers reaches around 25% in the research and the other brand added together only attain the remaining 25%. The further analysis will be presented below.
4.0 Data analysis
4.1 One way ANOVA analysis
4.1.1 Price versus Packaging
Hypothesis
H0= packaging does not affect the price of milk
H1= packaging affects the price of milk

Figure 11. One-way ANOVA: Price versus Packaging
According to the Figure 11 which shows that a relationship exists between the factor ‘packaging’ and the response ‘price’. Because the P-value is 0.000, which means that there is very strong evidence to suggest that packaging affects the price of milk and it also totally against the null hypothesis. In addition, there are 90 products under consideration are plastic packaging far more than the number of paper packaging products. Furthermore, it is obvious to see that the mean value of the price for paper packaging is $2.361 smaller than the plastic ones’ $3.181. And the R-Sq value is 10.63%, which means only 10.63% of the variation in price estimates is interpreted by packaging.

Figure 12. Residual plots for price
The residual plot at above showed the relationship between price and packaging. Therefore, the assumption for normality is reasonable. The fitted values for the plot are randomly distributed around 0, which does not indicate any problems.

Figure 13. Boxplot of price
The boxplot of price in Figure 13 shows the distribution of the price about in different packaging. According to compare, plastic packaging has the higher price than paper package and paper package. Plastics package has a wide range of the price and the price of is concentrate on $2 -$3. Because of paper package usually use on small size of milk and plastic package can use on large size, paper package may have lower price than plastic package.
4.1.2 Price versus Quality guarantee period
Hypothesis
H0= quality guarantee period does not affect the price of milk
H1= quality guarantee period affects the price of milk

Figure 14. Price versus quality guarantee period
It is easy to find out that the P-value in this test is 0.639, which indicated that there is insufficient evidence against null hypothesis. In another word to say that quality guarantee period does not conclusively affect the price of milk. In addition, the R-sq value of the test is 2.20%, which illustrate that just 2.20% of the variation in price is explained by the quality guarantee period. There are two data saw from the Figure 14 which state that the quality guarantee period of milk is two days and the mean price is $2.605 lower than others, which is only a rare evidence to show that the data of quality guarantee period influence the price of milk.

Figure 15. Residual plots of price
The residual plots for price show the relationship between price and quality guarantee period. The normal probability plot trend stable straight except for the biggest value which at 2.8on residual. The plots of versus fitted value are randomly distributed in the plot and do not indicate any problems in relationship.

Figure 16. Boxplot of price
The figure 16 at the above shows the boxplot of the price show the distribution of the price about in different quality guarantee period. 2 days left milk has the lowest price in all of milk. Every store is trying to sale milk that is nearly the quality guarantee date. The 9 days has the highest price in all milk. This milk was the last batch in the store and it is fresher than others. But the most 12 days milk is use the paper package and this milk may have lower price than fresh milk. So 12 days milk is not the most expensive milk in all of the milk. These plots may suggest that there has strong relationship between the quality guarantee period and price.
4.1.3 Unit price versus Type
Hypothesis
H0= the type of milk does not affect the unit price of milk
H1= the type of milk affects the unit price of milk

Figure 17. Unit price versus type
The P-value 0.022 showed above in the Figure 17 illustrate that there is strong evidence to suggest that the types of milk really affect the unit price of milk. Besides, the R-sq value is 10.96% of the variation in unit price and it is presented by the type of milk. The information at the bottom clearly indicated that different type of milk has distinct unit price, the highest value $3.388 of ‘others’ type of milk is about one more dollar than the lowest value of ‘fat free’ milk.

Figure 18. Residual plots for unit price
There are four small graphs in figure 18, they have shown the relationship between brand and unit price. The first small graph is normal probability plot and similar with straight line, second graph shows that all of points are distributed around 0 for random, third and fourth graph use two different types show the relationship, they cannot indicate any problems.

Figure 19. Boxplot of unit price
This plot is boxplot of unit price and it shows that the relationship between unit price and type, six milk types are distributed in figure 19. The median value of others is the largest than other types, moreover, full cream, low fat and new have almost same median, fat free has the smallest median. Therefore, through this plot can indicate a good relationship between unit price and type.
4.1.4 Unit price versus Brand
Hypothesis
H0= brand does not affect the unit price of milk
H1= brand affects the unit price of milk

Figure 20. Unit price versus brand
As shown above in Figure 20, the P-value of the test is 0.000 which represent there is very strong evidence to suggest that the brand affects the unit price without doubt. In addition, on the basic of R-sq value, about 24.72% of the unit price is illustrated by the brand.

Figure 21. Residual plots of unit price
There are four small graphs in figure 21, they have shown the relationship between brand and unit price. The first small graph is normal probability plot and similar with straight line, second graph shows that all of points are distributed around 0 for random, third and fourth graph use two different types show the relationship, they cannot indicate any problems.

Figure 22. Boxplot of unit price
This boxplot shows that the relationship between unit price and brand, there are 10 types of brands are displayed in figure 22. Pure organic has the highest unit price in all of brands and it has the highest median, in addition, Dairy farmers, Pauls and Queensland they are similar median, the median for Coles and Woolworths are lowest into these brands, Pauls and Queensland as the similar widely spread more than other brands. So this boxplot can indicate that they have very closed relationship between brand and unit price.
4.2 Two way ANOVA analysis
4.2.1 Price versus Quality guarantee period and Type

Figure 23. Price versus type, quality guarantee
There are some different feature in this data. The p-value of the type is 0.840 and it shows that there is insufficient evidence to suggesrt that the type can effect the price of the milk. However the p-value of the quality guarantee period is 0.987 and it show that there is no evidence to suggest that the quality guarantee period can effect the price of milk. Because of there are some other factor can influence the price of milk, quality guarantee period may not influence the price too much and only the date are near the quality guarantee date, the milk price may get the discount. Also, R-Sq is shown as 0.00% in this data. It means there quality guarantee period can not influence the price of milk.

Figure 24. Residual plots of price
As the figure 24 above shows, in the normal probability plot the most of points are follow the straight line and it prove the assumption for normality is reasonable. The plot of the residuals versus fitted value does not indicate any problems. It is fairly randomly scattered around 0.

4.2.2 Unit price versus Store and Brand

Figure 25. Unit price versus store, brand
There are some information about unit price versus store and brand using general linear model, it is showed in figure 25. The variables for this graph are store and brand. At first, store has p-value which is 0.000, it means there is very strong evidence which can indicate the relationship between store and unit price, so store has effects for mean unit price. Next, brand also has p-value which is 0.000, it means there is very strong evidence which can indicate the relationship between brand and unit price, thus, brand has effects for mean unit price. Therefore, store and brand not only can affect the mean unit price but also have interaction each other.
In addition, the R-Sq(adj) is 35.17% and it means there are 35.17% which can be indicate to store and brand in unit price.

Figure 26. Residual plots for unit price
Follow to this graph which explains the relationship with unit price, store and brand in figure 26.

From normal probability plot can get almost points in the straight line, and then in graph of versus fits all of points are distributed for random around 0, therefore, fitted value cannot indicate any problems.

4.3 Regression
Regression Analysis: Quality guarantee period vs Unit price and Container size

Figure 27. Regression analysis of Quality guarantee period vs Unit price and Container size
Hypothesis
H0= Quality guarantee period is independent of the unit price and the container size.
H1= Quality guarantee period depends on the unit price and the container size.

The regression analysis of Quality guarantee period versus Unit price & Container size, the relationship between Unit price and Container size combine together to affect the predictors of Quality guarantee period. In addition, the P-value of unit price and the container size both greater than 0.5 therefore there is insufficient evidence to suggest that the quality guarantee period depends on the unit price and the container size. Furthermore, the P-value 0.821 of the regression analysis also bigger than 0.5, hence it can support the null hypothesis as well. However, the R-sq value is too small so that it cannot present in a liner relationship between these variables.

Figure 28. Residual plots for quality guarantee period
These four plots in Figure 28 indicated the same meaning but using different ways. The plot of standardized residual showed the data was randomly scattered and no curvature exist in the relationship while the normal probability plot was unreasonably linear despite some unusual observations. Thus, the assumption of normality is not reasonable.
5.0 Conclusion
All in all, the further analysis makes several conclusions that the packaging really affects the price of milk while the quality guarantee period does not influence the price as much as the packaging does. In addition, the type and brand of milk are proved to have strong evidence to impact the unit price of milk by the one way ANOVA test. Furthermore, according to the results of two way ANOVA test, although quality guarantee period does not directly affect the changing of price, if combine quality guarantee period and the type of milk, it immediately influence the price. It shows that the fat free milk with longest quality guarantee period reaches the relative low price which means price inverse with the quality guarantee period but it depends on the how much the fat the milk has, more fat more expensive. Moreover, the interaction between store and brand of milk has strong evidence to impact the unit price of milk. However, some brand like Seven-Eleven which is not as popular as the other brand but in the retail shop the price often higher than supermarket. As a result, the statistical test showed clearly of the variables relationship and affect to each other.
Reference
Food outlook – Global market analysis. Food and agriculture organization of the United Nations. May 2012. p. 8, 54-54. http://www.fao.org/docrep/015/al989e/al989e00.pdf Appendix
Appendix A

Appendix B

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