The data shown in the table below represents the weight, in pounds, of a little girl, recorded each year on her birthday.
Age (in years) Weight (in pounds)
2 32
6 47
7 51
4 40
5 43
3 38
8 60
1 23
Part A: Create a scatter plot to represent the data shown above. Sketch the line of best fit. Label at least three specific points on the graph that the line passes through. Also, include the correct labels on the x-axis and y-axis.
Part B: Algebraically write the equation of the best fit line in the slope-intercept form. Include all of your calculations in your final answer.
Part C: Use the equation for the line of best fit to approximate the weight of the little girl at the age of 14 years old.
Part A: To create a scatter plot, start by labeling the x-axis with "Age (in years)" and the y-axis with "Weight (in pounds)". Then, plot each data point from the table on the graph. Here is the scatter plot:
(Please imagine a scatter plot with points representing the data)
To sketch the line of best fit, visually estimate a line that goes through the middle of the data points and appears to represent the overall trend. Make sure the line is balanced, with roughly equal numbers of points above and below it. Here is a rough estimation of the line of best fit:
(Please imagine a sketch of the line of best fit)
Label at least three specific points on the graph that the line passes through to verify its accuracy. For example, the line could pass through the points (2, 32), (5, 43), and (8, 60).
Part B: To write the equation of the best fit line in slope-intercept form (y = mx + b), we need to calculate the slope (m) and the y-intercept (b).
To calculate the slope, we use the formula:
m = (sum((x - mean(x)) * (y - mean(y)))) / (sum((x - mean(x))^2))
Let's calculate the mean of the x-values and y-values first:
mean(x) = (2 + 6 + 7 + 4 + 5 + 3 + 8 + 1) / 8 = 4.875
mean(y) = (32 + 47 + 51 + 40 + 43 + 38 + 60 + 23) / 8 = 42.125
Next, calculate the sum of the products of (x - mean(x)) and (y - mean(y)), and the sum of the squares of (x - mean(x)):
sum((x - mean(x)) * (y - mean(y))) = (2-4.875)*(32-42.125) + (6-4.875)*(47-42.125) + (7-4.875)*(51-42.125) + (4-4.875)*(40-42.125) + (5-4.875)*(43-42.125) + (3-4.875)*(38-42.125) + (8-4.875)*(60-42.125) + (1-4.875)*(23-42.125) = -76.25
sum((x - mean(x))^2) = (2-4.875)^2 + (6-4.875)^2 + (7-4.875)^2 + (4-4.875)^2 + (5-4.875)^2 + (3-4.875)^2 + (8-4.875)^2 + (1-4.875)^2 = 50.375
Now we can calculate the slope:
m = (-76.25) / 50.375 ≈ -1.515
To calculate the y-intercept, we use the formula:
b = mean(y) - (m * mean(x))
b = 42.125 - (-1.515 * 4.875) ≈ 49.532
Therefore, the equation of the best fit line in slope-intercept form is:
y = -1.515x + 49.532
Part C: To approximate the weight of the little girl at the age of 14 years old, we can substitute x = 14 into the equation of the best fit line and solve for y:
y = -1.515(14) + 49.532
y ≈ 29.808
Therefore, the weight of the little girl at the age of 14 years old can be approximated as 29.808 pounds.