Question

(1 point)
the table below shows my weight for the first few days after i was born. let w represent my weight, in ounces, at t days old.
demos protip: did you know that you can copy and paste this table into desmos?
a) use technology and all the data points to calculate the regression (\best fit\) linear model for w. round your result to the nearest hundredth.
w =
b) if i could leave the hospital when i weighed 5 pounds (80 ounces), according to your model, on what day was i allowed to go home (round to the nearest whole number)?
the model predicts that i was allowed to go home on day
c) i really went home on day 24. according to your model, how much did i weigh on day 24 (to the nearest whole ounce)?
the model predicts that on day 24 i weighed ounces.
note: you can earn partial credit on this problem.

Explanation:

Step1: Use linear - regression formula

Using a calculator or software (like Desmos) with the data points \((t_1,W_1)=(1,48),(t_2,W_2)=(4,55),(t_3,W_3)=(5,61),(t_4,W_4)=(8,69),(t_5,W_5)=(9,70),(t_6,W_6)=(10,70)\), the general form of a linear regression model is \(W = at + b\). Let \(x=t\) and \(y = W\). The formulas for \(a\) and \(b\) are \(a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\bar{y}-a\bar{x}\), where \(n = 6\), \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}\). Or using software, we get \(W=2.93t + 44.57\).

Step2: Solve for \(t\) when \(W = 80\)

Set \(W = 80\) in the equation \(W=2.93t + 44.57\). Then \(80=2.93t + 44.57\). Subtract 44.57 from both sides: \(80 - 44.57=2.93t\), so \(35.43 = 2.93t\). Solve for \(t\): \(t=\frac{35.43}{2.93}\approx12.09\approx12\).

Step3: Solve for \(W\) when \(t = 24\)

Substitute \(t = 24\) into the equation \(W=2.93t + 44.57\). Then \(W=2.93\times24+44.57=70.32 + 44.57=114.89\approx115\).

Answer:

a) \(W = 2.93t+44.57\)
b) 12
c) 115