Crunching Numbers: Making School Lunches Affordable and Delicious

Team 2825

Summary

The goal of this report is to assess the current school lunch program as it relates to the caloric needs of students and the cost to schools. Since the institution of the Healthy, Hunger-Free Kids Act in 2010, there have been significant changes in the dynamics of the school lunch programs. Because of this change, students are reluctant to buy the new healthy school lunches, the restrictions to certain food groups are depriving students of necessary calories, and the lunches are more costly to schools [1].

In order to fully understand how effective the Nutrition Standards in the National School Lunch Program are, it was necessary to determine the healthy caloric intake of individual high school students and how many are satiated by the lunches provided by the schools. First, our firm researched the various factors that affect caloric need, to create a model that can be used to find how many calories each individual student should intake at lunch. The factors that the model addresses include a student’s height, weight, age, gender, level of activity, and whether or not the student has eaten a healthy breakfast. The Institute of Medicine provides a model that calculates the Estimated Energy Requirement (EER) based on a student’s gender, age, height and weight [7]. We modified this model to accommodate the impact that the presence or lack of breakfast can have on EER. The result was a multivariable equation that can be individualized to determine each student’s caloric need based on their attributes.

Using this model and data from a random sampling of high school students, it was possible to create a set of graphs that showed how effective and average school lunch is at satisfying the caloric needs of an average student. The standard lunch, following the 2010 Act, contains up to 850 calories, which is not enough to healthily satiate the majority of growing teenagers [2] In fact, our firm’s conclusions supported the conclusion that only 8% of students were satisfied by the average school lunch and that a higher calorie meal was necessary to meet their needs.

Clearly, the data showed that a change in the food program was necessary. However, the schools have a staunch budget when it comes to school lunch spending. The sample school we created a model for, Underton High School, has a budget of about $7 per student per week. We assessed the school’s budget and the needs and wants of the students in order to create a model to reform the current school lunch program. The price of a lunch per student per day, $1.4947, was determined using government guidelines to have the correct proportion of the five major food groups, Grains, Vegetables, Fruit, Meat, and Dairy, and the average caloric intake needed (found by our equation) [13]. The price of a lunch per student per day, $1.4918, was also determined using student preferences of the five major food groups and the group of oils, fats, and sugars, and the average caloric intake needed [14]. We determined that only 60% of students in a school actually buy lunch, thus the money allotted for students that pack can be used to pay for other students’ lunches, resulting in a higher budget for each student. This important fact allows us to serve lunch that is under-budget and desirable to students, while also healthy for $1.4938 per student per day.

Table of Contents

Introduction......................................................................................................................................4

        Background..........................................................................................................................4

        Restatement of Problem.......................................................................................................4

        Global Assumptions.............................................................................................................5

Part 1: You are what you eat?..........................................................................................................5

Assessing the Question........................................................................................................5

Assumptions.........................................................................................................................5

        Designing the Model…………………………………………………………………......……5

        Testing the Model................................................................................................................7

        Strengths and Weaknesses...................................................................................................8

Part 2: One size doesn’t necessarily fit all.......................................................................................8

Assessing the Problem.........................................................................................................8

Assumptions.........................................................................................................................8

Designing the Model............................................................................................................9

        Fig. 1........................................................................................................................9

        Fig. 2......................................................................................................................10

        Fig. 3......................................................................................................................10

        Fig. 4......................................................................................................................10

        Fig. 5......................................................................................................................11

        Fig. 6......................................................................................................................11

        Fig. 7......................................................................................................................12

        Fig. 8......................................................................................................................12

        Fig. 9......................................................................................................................12

        Fig. 10....................................................................................................................12

Fig. 11....................................................................................................................12

Fig. 12....................................................................................................................13

Fig. 13....................................................................................................................13

Strengths and Weaknesses.................................................................................................14

Part 3: There’s no such thing as a free lunch.................................................................................14

Assessing the Problem.......................................................................................................14

Assumptions.......................................................................................................................14

Designing the Model..........................................................................................................15

Testing the Model..............................................................................................................16

Justification........................................................................................................................17

Strengths and Weaknesses.................................................................................................17

Conclusion.....................................................................................................................................18

Appendix........................................................................................................................................19

References......................................................................................................................................20

Introduction

Background

Since the dramatic rise in obesity in America within the past 20 years, combatting obesity has been an increasingly important issue to public safety [3]. Especially since 17% of children and adolescents are obese, it was necessary for the government to take action to ensure the health of American citizens. By this justification, the Healthy, Hunger-Free Kids Act was passed in 2010 to balance diets of school children. [1] An act heavily supported by First Lady Michelle Obama and passed by Congress, this new school lunch program offers restricted serving sizes and more nutritious food options [4]. Although the intentions are clearly aimed to protect the American population from obesity, this lunch program has received an unexpected reception from the public.

First, generally students do not take advantage of the healthy options that the school provides [1]. Students prefer sweeter, less healthy options to the healthy alternatives provided by the school. The new school lunch program did not take into consideration the tastes and preferences of the students, and it is clear from the choices of the students that they have not received the change positively. Most recent figures show that nearly one million students have opted out of the school lunch program [5]. This dramatic shift in the demand for school lunch has been detrimental to schools who are struggling to gain revenue from this more costly program.

Not only do students have a problem with the options provided by the school, but many of them claim that the restricted serving sizes do not meet their daily caloric needs. The USDA claims that the serving sizes are able to sustain an active 17 year old male, but the lack of choice and self-accountability of the students to make their own dietary decisions means that the school lunch program does not necessarily provide for every individual student [4]. An active 17 year old male has his own dietary needs, but they may even shift depending on what activity he does, if he eats a healthy breakfast, and other factors. Furthermore, two active 17 year old males may have extremely different dietary needs depending on their body types.  

The goal of a school is to provide meals that are able to satisfy the needs of its students. Economically, it is beneficial for a school to maintain a demand for its lunches. In order to ensure the success of the school lunch program in preventing obesity, the economic success of the school, and the well being of the students, the school lunch program must be reformed.

Restatement of Problem

The USDA has asked us to do the following:

1. Create a model that is able to take the attributes of an individual student and output the amount of calories that he or she should consume at lunch time.
2. Create a model that shows the distribution of high school students over the attributes that were assessed in the previous model.
3. Determine how many high school students have their caloric needs met by an average school lunch.
4. Create a lunch plan using mathematical modelling that stays within a school’s budget of $7 per student per week.
5. Determine what changes would be made to the lunch program if the school’s budget was decreased by $1 per student per week.

Global Assumptions

1. High School students get 50% of their calories from after school sources (snacks and dinner)[6].
2. High School Students get the other 50% of their caloric intake from breakfast and lunch combined (if breakfast is not eaten, all 50 percent are eaten at lunch).
3. Nationally, students will consume exactly the amount of calories they need per day [13].
4. All schools are following the Nutrition Standards in the National School Lunch Program.
5. We assume that no national food crises, such as mass drought, disease, or embargos will affect the price of food.

Part 1: You are what you eat?

Assessing the Question

        The goal of this section is to create a mathematical model that considers an individual’s attributes as inputs and outputs the amount of calories that they should consume at lunch, in order to be healthy. To ensure that the model we created was relevant, the inputs of our model are all factors that affect how many calories  an individual requires. Though many factors affect daily calorie usage, the most impactful factors that we delineated were related to a person’s normal routine. Sudden change in nightly sleep or exercise were not considered. In this respect, the model shows a person’s suggested caloric intake for lunch based on several factors: age, height, weight, gender, physical activity level, and whether or not a student eats breakfast.

Assumptions

1. Age, height, weight, gender, physical activity, and whether or not a student eats breakfast are the only factors that will impact caloric need at lunch. [7]
2. 50% of a student’s caloric intake for the day will come from a combination of breakfast and lunch, therefore the other 50% comes from food consumed after school hours (dinner, snacks, etc.) [8]
3. Breakfast counts as 20% of the daily caloric intake for students who eat it. [8].

Design of Model

To begin modelling this scenario, our firm found a way to relate all of the factors that  influence calorie usage to each other. Calories are a measure of energy, so all of the factors we took into account relate to energy. The Estimated Energy Requirement (EER) is the average dietary energy intake one requires to maintain energy balance in an individual of a normal weight. This requirement is dependent on age, gender, weight, height, and level of physical activity consistent [7]. Our firm’s model would be largely based upon this variable because we globally assumed that the amount of energy, in calories, that a person needed to consume would be consumed.

The EER predictive equations were developed by the Institute of Medicine Equation and were released in September 2002. The EER “equations were based on an extensive doubly labeled water database (considered the gold standard for Total Energy Expenditure measurement)” [7]. Total Energy Expenditure (TEE) is the sum of the basal metabolic rate, TEF (thermic effect of food), physical activity,” and other factors [7]. Our firm is reliant upon these scientific standards as we cannot conduct our own experiment to determine the behavior of calories.

Institute of Medicine’s EER Predictive Equation for Boys:

EER (kcal/d) = 88.5 - (61.9 x Age [years] + PA x {(26.7 x Wt [kg] + 903 x Ht [m])} +25

Institute of Medicine’s EER Predictive Equation for Girls

EER (kcal/d) = 135.3 - (30.8 x age [years]) + PA x {(10.0 x weight [kg]) + (934 x height [m])} + 25

These multi-variable equations will generate the caloric requirements a student requires based on their gender, height in metres, weight in kilograms, daily physical activity, and age in years. The physical activity coefficient (PA) is dependent on the activity level of the student.

[7].

In addition to age, height, weight, physical activity and gender, whether or not a student eats a nutritious breakfast has also been proven to impact the caloric requirements of a student’s school lunch.  In fact, students that skip breakfasts have been proven to experience weight gain, not loss [11]. Together, breakfast and lunch provide up to 50% of the caloric requirements of a high school student [8]. If a school lunch provides about 30% of one’s daily caloric needs, 20% of one’s daily caloric needs will be provided by breakfast [13]. Therefore, if one does not eat breakfast, their caloric needs from lunch increase.

Thus, the Institute of Medicine’s EER predictive equations can be slightly modified to accommodate the effect that a lack of breakfast can have on the daily lunch caloric requirement.

Modified Institute of Medicine’s EER Predictive Equation for Boys:

EER (kcal/day) = (0.5-.2(b)) (88.5 - (61.9 x Age [years] + PA x {(26.7 x Wt [kg] + 903 x Ht [m])} +25)

Modified Institute of Medicine’s EER Predictive Equation for Girls:

EER (kcal/day)=  (0.5-.2(b)) (135.3 - (30.8 x age [years]) + PA x { (10.0 x weight [kg]) + (934 x height [m]) } + 25)

Although these multivariable equations are very similar to the Institute of Medicine’s equations, the impact that a lack or presence of breakfast can have on the a student’s EER for lunch is represented by (0.5-0.2b) in which the variable b is 1 if a student eats breakfast and 0 if otherwise. Since both breakfast and lunch make up 50% of one’s caloric requirement, if one eats breakfast, lunch becomes 30% of one’s caloric requirement, as  USDA. If one does not eat breakfast, lunch becomes 50% of one’s caloric requirement.

Testing the Model

In order to test the model to see if it yields a reasonable number, a very active 18 year old female with a height of 1.65m and a weight of 56.69kg who did not eat breakfast was tested. The following calculations could be seen:

EER (kcal/day) for girls =  (0.5-.2(0)) (135.3 - (30.8 x age [18]) + PA[1.56] x { (10.0 x weight [56.69]) + (934 x height [1.65]) } + 25)

In total, this example shows that this female would need to eat a total of 1454.19 calories in a school lunch. This would make sense, because she is a very active young adult who would most likely need more than the generally accepted 2000 calories per day. Furthermore, since she did not eat a balanced breakfast, lunch becomes a much more important meal as far a receiving her daily calories. If this particular female, considering her personal attributes, consumed a lunch that is only 850 calories (as described in the School Lunch Guidelines[1]), this student would only receive approximately 58.4% of the calories that she should aim to consume in a lunch.

        We compared this female’s caloric need at lunch to that of a female with the same physical attributes, age, and exercise habits who does eat breakfast. The female student with a healthy breakfast will require only 872.514 calories for lunch.

        We also compared the first female student’s caloric need at lunch to that of a female with the same physical attributes, age, and breakfast habits who does not exercise. The sedentary female student will require only 856.95 calories for lunch.

Strengths and Weaknesses

        The model calculates the Estimated Energy Requirement (EER) for various attributes such as a student’s height, weight, age, gender and consumption of breakfast. This model depicts how two students with the same gender and stature could have different caloric needs if they do not eat breakfast. This fifth factor, if accommodated, would help any high school administration determine exactly how many children are not satiated by their standard school lunches.

        However, this model does not robustly determine the EER on a short term, day to day basis as it does not account for temporary factors like stress-induced eating and sleep deprivation. Stress, for example, causes overeating in 36% of people whereas 27% of people surveyed in the same random sample reported that they skipped a meal due to stress [17]. Sleep deprivation, on the other hand, has proven to cause adults to reduce their caloric intake of protein during lunch and increase their caloric intake of junk foods filled with carbohydrates and fats [18]. Since spells of sleep deprivation and stress are not long-term, these factors could not be accommodated into the model accurately.

Part 2: One size doesn’t necessarily fit all

Assessing the Problem

        Although the Healthy, Hunger-Free Kids Act sets a maximum calorie level of 850 for school lunches, many high school children are dissatisfied with this amount [2]. High schools students’ caloric needs range greatly based on height, weight, gender, physical activity, age, and whether or not they ate breakfast. For example, highly active students or those who have not eaten breakfast will require a significantly greater amount of sustenance from school lunch than sedentary students who ate a healthy breakfast. The goal of this section was to determine approximately what percent of high school students have their caloric needs met by standard school lunches, taking into account the variables we used in the model above. We decided to use census data to assess the distribution of these variables, and get an approximation of the range of caloric needs for high school students.

Assumptions

1. The amount of calories in a standard school lunch is 850 [2].
2. A random sample of the genders, ages, heights, and physical activities of 437 high school students from around the nation is representative of the same statistics for all U.S. high school students [13] (see Appendix).

Designing the Model

In order to get a range or distribution of caloric needs of United States high school students, we completed a random sample of 437 students of both genders from all states in the U.S. located on the Census at School resource [9]. This census information included height, gender, age, among other extraneous details about the randomly selected students. Because the census data did not include weights of the students in the sample, we assigned various weights to each of them based on the average weight distribution for their age. [3] (see Appendix). For example, the 3rd percentile for 14 year old female girls is 35 kg. If a hundred 14 year old girls are randomly sampled, 3 girls were randomly given a weight between 30 kg (0 percentile) and 35 kg (3rd percentile).

 From the census questionnaire, participants listed their favorite physical activity and the average amount they spend participating in outdoor activities. We used this data to assign each participant a level of physical activity coefficient (see physical activity coefficient requirements in part 1). After inputting height, weight, gender, and age into the equation, we created a set of caloric lunch needs in which all students went without breakfast for each gender, and a set in which all students ate breakfast for each gender.

Below are graphs that show where on the range of caloric need various students fall. In these graphs, one variable is the focus while the others are considered negligible. In this way, it is easy to see the correlation between specific variables and how they influence the caloric need of a student. Each point is representative of one of the 437 students surveyed in the census. First, we compared the lunchtime caloric needs of males and females who did not eat breakfast to those who did.

Figure 1: Lunchtime Caloric Need for High School Males who Do Not Eat Breakfast

In this graph, the trend shows that most males that do not eat breakfast must eat around 1600 calories at lunch. The numbers range from 1200-2300 calories that should be eaten at lunch time.

From this data, not one person who did not eat breakfast is satisfied by the 850 calorie limit for school lunches.

Figure 2: Lunchtime Caloric Need for High School Males who Eat Breakfast

In this graph, most of the male participants require 1000 calories at lunch, however, the numbers range from about 640 calories to 1500 calories. Although some students are satisfied by the 850 calorie limit, most of the participants require more than that to sustain themselves.

Figure 3: Lunchtime Caloric Need for High School Females who Do Not Eat Breakfast

This graph shows that females who do not eat breakfast need to intake a range of calories from about 700-1950. Although a few of these students are satisfied by the 850 calorie limit on school lunches, the majority of them are not reaching the amount of calories they need to maintain a normal weight.

Figure 4: Lunchtime Caloric Need for High School Females who Eat Breakfast

In this graph, the range of caloric intake that females need if they had breakfast ranged from about 410 calories to 1300 calories. The majority of these students are satisfied by by the 850 calorie per lunch limit.

As shown in the above graphs, the range of caloric needs for students that eat breakfast is significantly lower than those that do not. Furthermore, females generally require fewer calories than males. Only females who have eaten breakfast are likely to be satisfied by a lunch meal that is under 850 calories.

Next, we compared the daily caloric needs of male and female students based on their level of physical activity, using the physical activity coefficients described in the previous section. Keep in mind, these values no longer relate to the calories a student requires per lunch, but instead it relates to the calories a student requires per day. This is because we cannot determine whether or not a student has or has not eaten breakfast, so we cannot tell whether a student needs his or her lunch to represent 50% of his or her diet, or 30%.

Figure 5: Daily Caloric Needs for Sedentary Male Students

This graph shows that males who participate in no physical activity require a range of calories from about 2175 to 2825 in a day. Most of these students, however, require around 2400 calories in a day.

Figure 6: Daily Caloric Needs for Low Activity Male Students

In this graph, low activity males require a range of calories per day from about 2225 to 3475.

Figure 7: Daily Caloric Needs for Active Male Students

This graph shows that that active male students require a range of calories per day from about 2300 to 4100. However, most of these students require around 3300 calories.

Figure 8: Daily Caloric Needs for Very Active Male Students

This graph shows that very active males require a range of calories per day from a range of about 3200 calories until 4750 calories.

As shown among these graphs, as the level of physical activity increases among males, the calories they require also increases.

Figure 9: Daily Caloric Needs for Sedentary Female Students

This graph shows that females who participate in no physical activity require a range of calories from about 1460 to 1980 in a day.

Figure 10: Daily Caloric Needs for Low Activity Female Students

In this graph, low activity females require a range of calories per day from about 1740 to 2750.

Figure 11: Daily Caloric Needs for Active Female Students

This graph shows that that active female students require a range of calories per day from about 1690 to 2790. However, most of these students require around 2300 calories.

Figure 12: Daily Caloric Needs for Very Active Female Students

This graph shows that very active females require a range of calories per day from a range of about 1450 calories until 3525 calories. However, most students fall within a range of 2500- 3500 calories per day.

As shown among these graphs, as the level of physical activity increases among males, the calories they require also increases.

Finally, we used the statistic that 80% of students do not have breakfast and 20% of students do to estimate the lunchtime caloric need of all 437 students in the sample (http://share.kaiserpermanente.org/static/weightofthenation/docs/topics/WOTNCommActTopic_School%20Food_F.pdf). From the sample population, the amount of calories needed for lunch without breakfast was multiplied by 0.8 and the amount of calories needed for lunch with breakfast was multiplied by 0.2. This process yielded the data on the graph below.

Figure 13: Lunchtime Caloric Need for High School Students

This graph shows a compilation of all high school students and the calories calculated that they require in a school lunch. Depending on each student’s individual attributes, the range of calories that students need are between approximately 625 and 2150 calories. However, it is important to note that the vast majority of students require over 850 calories in a school lunch: proving that the standard meal size set by the Healthy, Hunger-Free Kids Act.

Based on the data from the census and the statistic that only 20% of children eat breakfast, 35 out of the 437 students require 850 calories or less, based on the models in Part 1. Therefore, only 8% of high school students receive a sufficient amount of calories.

Strengths and Weaknesses

        The strengths of modelling this data are clear: it is very easy to see how many students are satisfied by the 850 calorie limit or how much one single variable affects the calorie requirement for each student.

One of the weaknesses of this model is that the data was not always the most reliable. Although we started with 500 participants in the census, many of the responses had to be discarded because their answers were deemed unreliable. Also, because the model had multiple variables, it was not always possible to isolate one single variable as being responsible for the change in a person’s requirement for calories.

Part 3: There’s no such thing as a free lunch

Assessing the Problem

        The goal of this problem is to relate an ideal lunch plan to the real world and test its practicality. If Underton High School has a budget of $7 per student per week, what is the most effective way to address the student’s wants within the school’s ability to provide?

        This problem required us to assess what foods students favor, and what foods will most likely be bought by the students. As well, we took into consideration the most healthy and balanced lunch meal as far as food-group recommendations that could be bought within the budget. The result of our analysis would reveal a meal plan that is desirable, healthy, and within budget.

Assumptions

1. We assume that the five major food groups are Grains, Vegetables, Fruits, Meat and Dairy [13].
2. Assume all students need the same percentages of grains, vegetables, fruit, meat or protein, and dairy.
3. We assume that the school is able to pay for all factors needed to package, prepare and serve food, from materials to equipment to people.
4. The amount of money a school has to spend on food is equal to 1.4 dollars multiplied by the number of students in the school. This is found by dividing the given seven dollars per student each week by five days per week to give 1.4 dollars per student each day.
5. We assume that the number of serving sizes for a food group divided by the total number of servings for all food groups is directly related to the number of calories of that food group an individual should eat.
6. We assumed that the dairy products sold at lunches would be mainly milk; other dairy products are sold with other food groups (such as cheese on a cheeseburger) or sold individually, both in negligible amounts. Therefore, the favored percentage of dairy is based on the number of kids from the census data that chose milk as their favorite beverage.
7. Milk and water are the only two beverages available to students at lunch time.
8. We assume that the price of the food bought by the schools is the wholesale price, which is eleven thirtieths (11/30s) of the retail price [14].
9. We assume that if students get thirty percent of what they want food-group wise, they will be satisfied. Although there is no specific data to support this, what the students prefer is not particularly different from what  the USDA suggests [13]. Therefore we feel that the students gaining thirty percent of what they want will be enough for them to be satisfied.

Designing the Model

In order to design the model, we first determined the percent of calories that each major food group, Grains, Vegetables, Fruit, Meat, and Dairy, should take in a person’s diet daily. We used these proportions to determine the number of calories that a person should eat for each food group in a lunch. Then we used Nutrient Lists to find the average amount of calories per 100 grams of each food group [12]. Using dimensional analysis, we were able to find the average number of grams that an individual should consume for lunch. We found the average caloric intake needed, using our equation and the census of 437 students, was 1329.64 calories, which also took into account that only 20% of students eat breakfast [8]. This calorie value was used to find the number of calories of each food group that should be consumed. Thus the grams that should be consumed, leading to the overall price for a healthy lunch. It was also used to find the number of calories of each food group that are wanted or favored by high school students, thus the grams they want to consume, and the overall price for an idealized meal (if chosen based on student preferences). The prices for each food group were found by taking the U.S. city average retail food prices and finding the average price for each food group, then taking eleven thirtieths of the price to find the approximate wholesale price [15]. Finally, this caloric value was used in the final determination of how to fit the needs and wants of students within the budget given to our consulting firm.

Pneeded= Pgrains[Sgrains(C)(100 grams/279 calories)] + Pvegetables[Svegetables(C)(100 grams/63 calories)] + Pfruit[Sfruit(C)(100 grams/85 calories)] + Pprotein[Sprotein(C)(100 grams/216 calories)] + Pdairy[Sdairy(C)(100 grams/229 calories)]

        This equation finds the price of a school lunch per student per day. The Sfood group variables determine the proportion of calories that should be within a certain food group based on the amount of serving sizes needed each day (there are no units). C is the average number of calories a student needs to intake, based on the caloric intakes we found using our equation from Part I with data from the census. The 100 grams/X calories uses dimensional analysis to convert the calories in each food group into grams, which are easier to use when finding prices and buying in bulk (units are grams/calories). Finally, the Pfood group variables are the average price of food per grams within each food group. Sgrains=0.36, Svegetables=0.16, Sfruit=0.12, Sprotein=0.24, and Sdairy=0.12, using the serving sizes specified [14]. Pgrains= $0.00097/gram , Pvegetables= $0.0011/gram, Pfruit= $0.0014/gram, Pprotein= $0.0030/gram, Pdairy= $0.0036/gram, and Poil/fat/sugar= $0.0022/gram.

Pwanted= Pgrains[Wgrains(C)(100 grams/279 calories)] + Pvegetables[Wvegetables(C)(100 grams/63 calories)] + Pfruit[Wfruit(C)(100 grams/85 calories)] + Pprotein[Wprotein(C)(100 grams/216 calories)] + Pdairy[Wdairy(C)(100 grams/229 calories)] + Poil/fat/sugar[Woil/fat/sugar(C)(100 grams/448 calories)]

For explanations of Pfood group, C and 100 grams/X calories, see the above paragraph. Wfood group is the percentage of calories that high school students want to eat within each food group, based on data from the census.  Wgrains= 0.33, Wvegetables = 0.019, Wfruit = 0.081, Wprotein = 0.51, Woil/fat/sugar = 0.055, and Wdairy = 0.067 where dairy refers to the percent of people who chose milk as a beverage.

Pplan= Pgrains[Bgrains(C)(100 grams/279 calories)] + Pvegetables[Bvegetables(C)(100 grams/63 calories)] + Pfruit[Bfruit(C)(100 grams/85 calories)] + Pprotein[Bprotein(C)(100 grams/216 calories)] + Pdairy[Bdairy(C)(100 grams/229 calories)] + Poil/fat/sugar[Boil/fat/sugar(C)(100 grams/448 calories)]

Where Bfood group= (0.70)Pfood group+(0.30)Wfood group. This gives new proportional values for what percent of calories should be given to each food group. Bgrains= 0.351 , Bvegetables= 0.1177, Bfruit= 0.1083, Bprotein= 0.321, Bdairy= 0.1041, and Boil/fat/sugar= 0.0165. Pplan is the price of a lunch per student per day, considering nutrition guidelines and student preferences; this is the price Underton High School should expect to pay.

Testing the Model

The sample school, Underton High School, was used in this test. The high school of 436 students was given a strict budget. The plan gave the students $7 a week to buy school lunches. With this implemented, each student will have a $1.4 lunch. This means that the school has a budget of $610.40 per day to buy student lunches.

Pneeded=0.00097[0.36(1329.64)(100 grams/279 calories)] + 0.0011[0.16(1329.64)(100 grams/63 calories)] + 0.0014[0.12(1329.64)(100 grams/85 calories)] + 0.0030[0.24(1329.64)(100 grams/216 calories)] + 0.0036[0.12(1329.64)(100 grams/229 calories)]=$1.494718861

Pneeded=$1.4947 per lunch per student per day

This was based off the calories needed of a healthy diet.

The Pwanted was calculated the same way as the Pneeded by different values for what the healthy daily percentage of food in that group was replaced for the percent of what students want, not necessarily what is healthy for them.

Pwanted=$1.4918 per lunch per student per day.

Pplan=$1.4938 per lunch per student per day (based off of needs and wants)

Pneeded for entire school=$651.70 per day (all 436 students)

Pwanted for entire school=$650.44 per day (all 436 students)

Pplan for entire school=$651.32 per day (all 436 students)

The price needed is higher than the price wanted, but only by a miniscule amount.

Although a specific sensitivity analysis was not run, the robustness of our model is evident in how little the price per lunch per student per day changes as the food group caloric proportions are manipulated (Sfood group, Wfood group, Bfood group). The value changes from $1.4947 (with S) to $1.4918 (with W) to $1.4938 (with B), therefore the model is accurate to two decimal changes.

Justification

        To test our model, we used data from the census from one of the sources given to us. We felt that this accurately portrayed a cross section of high schools all over the United States, because we chose to use data from all state, male and female, for ninth, tenth, eleventh and twelfth grade, from 2010 to 2013. Therefore, we used the census as a hypothetical “school” of four hundred and thirty seven students to test the price.

        The use of Bfood group to determine the budget price is justifiable because the new proportion for each food created puts more emphasis on healthy eating (70 percent) than what the students want, yet also takes into account what students prefer (30 percent). Although these numbers were arbitrary, we felt that they would show the school’s focus on health and nutrition while also appeases the students.

Strengths and Weaknesses

Though our model addressed many facets, there are some ways in which it is strong and others in which it is weak. It is strong because it assesses the cost of lunch per student per day, giving the school compact data that is easy to understand. Also, because the price is determined mainly by the caloric intake, it is easy to use and understand by a range of people. This is a drawback as well because it does not account for varying food prices that could be affected by geographic location, rural versus city location, and other environmental factors. The change in price could be easily reflected in our model by changing the average price for a food group per gram. Another weakness is the result that our price is under budget is dependent on the fact that only 60% of students buy lunches every day. In different geographic and socioeconomic environments, this percentage might change, thus changing the amount that can be spent on each student.

Conclusion

        The first question that we assessed in this report was how to model an equation that could take a person’s attributes and calculate how many calories they should eat in a lunch. Then, we graphed this data on a distribution curve to show how many people from a sample would need a certain amount of calories depending on their specific attributes. From a set of 12 graphs, we determined that only 8% of students get the amount of calories that they need from a standard school lunch. 92% of students, for various reasons, require more than the legal limit of 850 calories per lunch.

        Finally, we developed a model that found the average price per lunch per person per day, based on the proportion of average calories eaten by an individual at lunch (found with our equation from Part I)

The price of food for what the students need exceeds the price of the school budget. However, it is found on average that only 60% of students buy school lunches [16]. So, with the same budget given in the proposal ($7 a week per student, $610.40 per week per schoool), the school only has to support about 262 students. This means that the budget the school has in place for each student that buys lunch is actually $2.33 per day per student for a school lunch. Therefore, there is a lot of extra money in place for a student meal. This allows the students to order the food that they want and prefer, and the school can obtain healthy choices.

Also this model, which assumes that on average only 60% of students buy school lunches a day, the price of paying for a student’s lunch can be reduced to $1.2 a day, or $6 a week per student. If the budget is reduced, then the amount of money needed for one school day at the sample school is $523.20. With the 60% assumption, only 262 students will need to buy a lunch. Therefore, the budget per person for a day is $1.9969, which rounds up to $2.00. This still is 50 cents more money spent on food than is needed to satisfy student caloric needs, student preferences, and the final plan.

Furthermore, if Underton High School, wanted to cut school lunch funds even more, it could while still being within the given budget and pleasing the students. With the Pplan=$1.4938 and the fact that the school only needs to provide 262 students, the bare minimum budget for the entire school for one day is $391.39, as compared to the original $610.40; this is only 64% of the suggested budget. However, this budget should be used with caution because assuming that exactly 262 students will buy lunch on any given day could result in lack of food if more students buy that day. Therefore, our consulting firm suggests that Underton and other high school use this budget to buy the different proportions of food groups, therefore ensuring healthy and nutritious meals as well as student satisfaction, but that they do not buy for only 60% of their students, instead adjusting to how many of their students buy daily.

Appendix:

Below is a chart of the first 20 entries from our random sample of 437 students. This serves as an example of how we gathered and interpreted the data. All data except for the Weight and Calorie Intake data was gathered from the census. The weight, as described above in Part 2, was randomized from distribution graphs and the calorie intake data was calculated using the model in Part 1. The favorite food and beverage information was used in Part 3.

References

1. http://healthland.time.com/2013/08/29/why-some-schools-are-saying-no-thanks-to-the-school-lunch-program/
2. http://www.huffingtonpost.com/2012/09/26/michelle-obamas-low-calorie-school-lunch-video_n_1914394.html
3. http://www.cdc.gov/obesity/data/facts.html
4. http://cspinet.org/new/pdf/new-school-meals-faq.pdf
5. http://www.washingtontimes.com/news/2014/mar/6/1m-kids-stop-school-lunch-due-michelle-obamas-stan/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed:+Nation-TheWashingtonTimesAmericasNewspaper+(Nation/Politics+-+The+Washington+Times)
6. http://www.healthyschoolfoodsnow.org/the-state-of-school-nutrition/
7. http://www.globalrph.com/estimated_energy_requirement.htm
8. http://share.kaiserpermanente.org/static/weightofthenation/docs/topics/WOTNCommActTopic_School%20Food_F.pdf
9. http://www.amstat.org/censusatschool/RandomSampleForm.cfm
10. http://www.cdc.gov/growthcharts/charts.htm#Set1
11. http://www.healthychildren.org/English/healthy-living/nutrition/pages/The-Case-for-Eating-Breakfast.aspx
12. ttp://ndb.nal.usda.gov/ndb/nutrients/index
13. http://www.cnpp.usda.gov/Publications/MyPyramid/OriginalFoodGuidePyramids/FGP/FGPPamphlet.pdf
14. http://www.thedesigntrust.co.uk/what-price-to-use-when-talking-to-retailers/
15. http://www.bls.gov/cpi/cpid1401.pdf
16. http://www.princeton.edu/futureofchildren/publications/journals/article/index.xml?journalid=36&articleid=98&sectionid=608
17. http://www.apa.org/news/press/releases/stress/2012/impact.aspx?item=2
18. http://www.apa.org/news/press/releases/stress/2012/impact.aspx?item=2