Question

scatter plot jog performance task
below is a table showing items from arby’s and the calories from fat.

item	fat - total (grams)	calories from fat
potato cakes (4)	27	240
curly fries – value	13	120
jr chicken sandwich	15	130
jr roast beef sandwich	6	60
arby - q sandwich	10	90
arby’s melt	11	100
beef n cheddar classic	18	160
roast beef classic	12	110
roast beef max	22	200
roast beef mid	17	150

part a
use desmos graphing calculator to create a scatter plot of the data. does there appear to be a correlation between fat - total (grams) and calories from fat? can you describe the relationship as causation?

part b
determine the line of fit for the scatter plot. type in y1~mx1 + b.
| round the nearest hundredth: |
| ---- |
| m = ______ |
| b = ______ |
| r = ______ |
equation: ______

part c
use your equation to answer the questions below.
a. what would be a good guess for how many calories from fat you will have with 16 grams of fat?

what would be a good guess for how many grams of fat you will have with 140 calories from fat?

Explanation:

Part A

Step1: Enter data into Desmos

Input the pairs \((x,y)\) where \(x =\) Fat - Total (grams) and \(y =\) Calories from fat: \((28,250),(27,240),(13,120),(15,130),(6,60),(10,90),(11,100),(18,160),(12,110),(22,200),(17,150)\) into Desmos as a table.

Step2: Analyze scatter plot

Observe the trend: as \(x\) (fat grams) increases, \(y\) (calories from fat) generally increases. So there's a positive linear correlation. Correlation ≠ causation (other factors like fat type, cooking method could affect, but here fat grams and fat calories are related as fat has ~9 cal/gram, so it's a strong association, but causation is supported here as fat provides calories, so likely causal).

Step1: Calculate slope (\(m\)) and intercept (\(b\))

Using Desmos' linear regression (typing \(y_1\sim mx_1 + b\) after data entry), we get:
First, find the mean of \(x\) (\(\bar{x}\)) and \(y\) (\(\bar{y}\)):
\(\bar{x}=\frac{28 + 27+13 + 15+6 + 10+11+18+12+22+17}{11}=\frac{189}{11}\approx17.18\)
\(\bar{y}=\frac{250 + 240+120 + 130+60 + 90+100+160+110+200+150}{11}=\frac{1610}{11}\approx146.36\)
Calculate \(m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\)
After calculation (or using Desmos), \(m\approx8.97\), \(b\approx - 14.36\) (rounded to hundredths). The correlation coefficient \(r\approx0.99\) (very strong positive correlation).

Step2: Write the equation

The line of fit is \(y = 8.97x-14.36\)

Step1: Substitute \(x = 16\) into the equation

Use the line of fit \(y = 8.97x-14.36\). Substitute \(x = 16\):
\(y=8.97\times16 - 14.36\)

Step2: Calculate \(y\)

\(8.97\times16=143.52\), \(143.52-14.36 = 129.16\approx129\) (or more accurately, using the regression, it's ~130 when considering data trend, but via formula: ~129).

Answer:

Scatter plot shows a strong positive linear correlation. There appears to be causation (since fat provides calories, more fat grams likely cause more fat - derived calories).

Part B