I am trying to find the line of best fit and this makes no sense at all.
I have a table - here are the values for x, length, and corresponding values for weight, y
60 1-5
62 114
64 124
66 131
68 139
70 149
72 158
I'm asked to find the line of best fit and round to hundredths.
When I do that, I come up with: y = 4.36x - 256. 14
Then, the question asks how much I would expect something that is 75 in length to weigh.
So, I do: y = 4.36(75) - 256.14 and get 70.86
This makes absolutely no sense based on the table!
Where have I made my mistake?
Thank you.
To find the line of best fit, you need to perform linear regression on the data points. It seems you have made an error in your calculations. Let's go step by step to ensure accuracy.
Step 1: Calculate the mean (average) of x and y values.
- Mean (x) = (60 + 62 + 64 + 66 + 68 + 70 + 72) / 7 = 66.
- Mean (y) = (114 + 124 + 131 + 139 + 149 + 158) / 6 = 134.83 (Note: it seems there is a mistake in the table; there is no corresponding y value for x = 60).
Step 2: Calculate the difference between each x value and the mean of x (66), and the difference between each y value and the mean of y (134.83). Let's call these differences dx and dy.
- dx = x - Mean (x):
60 - 66 = -6,
62 - 66 = -4,
64 - 66 = -2,
66 - 66 = 0,
68 - 66 = 2,
70 - 66 = 4,
72 - 66 = 6.
- dy = y - Mean (y):
114 - 134.83 = -20.83,
124 - 134.83 = -10.83,
131 - 134.83 = -3.83,
139 - 134.83 = 4.17,
149 - 134.83 = 14.17,
158 - 134.83 = 23.17.
Step 3: Calculate the product of dx and dy, as well as the square of dx.
- dx × dy:
(-6) × (-20.83) = 125.00,
(-4) × (-10.83) = 43.32,
(-2) × (-3.83) = 7.66,
0 × 4.17 = 0.00,
2 × 14.17 = 28.34,
4 × 23.17 = 92.68,
6 × 23.17 = 139.02.
- dx^2:
(-6)^2 = 36,
(-4)^2 = 16,
(-2)^2 = 4,
0^2 = 0,
2^2 = 4,
4^2 = 16,
6^2 = 36.
Step 4: Calculate the sums: Σ(dx × dy) and Σ(dx^2).
- Σ(dx × dy) = 125.00 + 43.32 + 7.66 + 0.00 + 28.34 + 92.68 + 139.02 = 436.02.
- Σ(dx^2) = 36 + 16 + 4 + 0 + 16 + 16 + 36 = 124.
Step 5: Calculate the slope (m) and the y-intercept (c) using the formulas:
- m = Σ(dx × dy) / Σ(dx^2) = 436.02 / 124 = 3.52 (rounded to two decimal places).
- c = Mean (y) - (m * Mean (x)) = 134.83 - 3.52 * 66 = -87.48.
So, the correct equation for the line of best fit is y = 3.52x - 87.48 (rounded to two decimal places).
Now, let's calculate the expected weight for an x value of 75:
y = 3.52 * 75 - 87.48 = 232.52 - 87.48 = 145.04 (rounded to two decimal places).
Therefore, if the length is 75, we would expect it to weigh approximately 145.04.