Answer

Curve-fitting Project – Linear Model

This project is based on data from MyFoodDiary, a site that provides nutritional information (retrieved from: https://www.myfooddiary.com/foods/search?q=cheese+pizza+12%22). This is based on nutritional facts of Pizza, based on nutritional content and value. While a slice of pizza may not be equal for the different brands sampled in the dataset, the data points express “Fat Content” in grams, this measures specifically the total fat content of the pizza in a single serving, yet this is mostly made of the saturated fat, the nutritionally dangerous component. Moreover, the fat content can be expressed in percentage, a proportion of the whole compared to other nutritional contents of the slice of pizza but this percentage measure is not considered in the linear modelling. Further, the calorie count in the slices of pizza is also recorded. A calorie is a unit that is used to measure energy. When excess counts of calories are ingested by humans, it is likely to lead to weight gain and more serious lifestyle problems like diabetes and so it necessitates that the calorie measure is observed by food consumers (Foster, et al., 2010).

In this case, cheese pizza is the fast-food picked for assessment. As prevalently loved, pizza’s high fat content is hypothetically connected to high calories. This is based on the scientific fact that fats are the greatest providers of calories as compared to carbohydrates and proteins, as also articulated by N. H. S. (2013). Therefore, the high fat content of cheese pizzas are likely to lead to high calorie count. Therefore, a linear model is formulated to aid in predicting the calorie count for each pizza serving to enhance lifestyle watch for pizza lovers. This is intricately connected with the knowledge that 2,000 calories a day is used for general nutrition advice. But it has generally become increasingly difficult to trust the food manufacturers with their information on the foods as consumers become more responsible and concerned based on the exponential progression of lifestyle diseases and complications (Nestle, 2013). Therefore, this model is a helpful tool for enhancing food consumer responsibility.

Fig 1: Scatter plot of Calories and Fat Content

The scatter plot of the data points indicate linear progression in an increasing direction. This is further confirmed by the best line of fit as seen below.

Fig 2: Scatter plot of Calories and Fat Content with line of best fit.

This line of best fit indicates that with the progression of fat content of Cheese Pizza, there is an integral progress in calorie count. The line of best fit shows us what would be in the data points if there was no random error. In this sense, an increase in fat content in the cheese pizzas leads to an increase in the level of calories. In other words, brands with high fat content will most likely have high calories. This plot indicates that about 83.1% of the variation in Calories as the dependent variable is explained by the model that as Fat Content as the independent variable. The equation of the regression line is y = 14.699x + 98.515.

Determining the Slope of the Line

The coordinates chosen for determining the slope of the curve are (5, 172.01) and (20, 392.495).

m = (y₂ – y₁)/(x₂-x₁)

= (392.495 – 172.01)/(20-5)

= 14.699

Generally, the slope is 14.699

The slope or coordinate of geometry is the measure of the steepness of the line of best fit, or the regression line. It expresses the change in y for a unit change in x along the line. It represents the rate in change in the predicted variable as the independent variable changes. In this case, the slope (14.699) describes the predicted values of Calories given values of Fat Content in the Cheese Pizzas.

Coefficients of Determination and Correlation

R² = 0.831

r = 0.9116

The high coefficient of determination (R²), which is close to 1, indicates that the line is a good fit for the data.

The data exhibits a very strong positive correlation between fat content and calories (0.912). 

There is a very strong linear relationship between Fat Content and Calories, this indicates that as Fat Contents increase in the various Cheese Pizza brands, the Calories count will also increase highly and voce-versa.

Prediction

The equation of the linear model formulated is y = 14.699x + 98.515

Where y is the Calories and

x denotes the Fat Content.

In this case, to predict the level of calories of a brand of cheese pizza slice that has a fat content of 29g in a serving, the results are as follows:

 y = 14.699x + 98.515

                                   = 14.699×29 + 98.515

                                   = 524.786

Therefore, it is predicted from the linear model that a slice of cheese pizza with 29g fat content will provide 524.8 calories.

Discussion

The linear regression model that fits the data is amazingly interesting in the sense that it explains a large part of the variation in the predicted variable, and that the best line of fit is significantly a good fit for the data. The data points do not only express meaningful values, but significant. Therefore, the model is generally good, fitting use for prediction with warranting significance.

The scatter plot with the best line of fit is particularly interesting as it indicates that there is a possibility of several lines of fit for the data, but the best line of fit, which is the linear model in this case, caters for up to 83.1% of the variation in the dependent variable – Calories. The line is determined to be a good fit based on the coefficient that is equal to 1.

Moreover, there is a confirmation of the data that there cannot be a regression model unless there is correlation between the dependent and independent variables. The very strong correlation between Fat Content and Calories variables validate the model and its strength in prediction. The formulated model, then, can be to predict fat content value by use of the regression technique while assuming the occurrences of random errors that are not catered for in the best line of fit.

The strong positive correlation finding in this project is highly consistent with expectations based on nutrition science, a gem which also forms part of the anchorage in the project by Popkin, Adair, and Ng (2012). In a society that is faced with a rising menace of obesity and general lifestyle diseases, the necessity to examine and assess contents of ingested foods escalates and this linear model will be very timely in the paradigm shift. Therefore, the meaningfulness of the model is unquestionable.

This linear model can be used by both food manufacturer and consumer. The former in determining the right levels of fats to include in the cheese pizzas for marketability of brands and for social responsibility of production, while the latter can find meaningfulness in this model in choosing the best brands basing on the Calorie question.