# Linear Regression Questions - Suspected Typo?

My sister just submitted an assignment and got a few questions marked incorrect (electronically) but I've just checked over them and don't believe this to be the case. Can someone either point out where I'm going wrong or confirm my suspicions please?

Here is the data (sorry for the formatting!):

Running Water Temperature (oC),Elevation (masl)

10.6,2565

11.2,2591

10.4,2595

6.9,3263

8.3,3411

1.2,4508

1.4,4468

5.1,3747

5.5,3556

7,3499

9.8,2931

14.9,1551

18.1,1095

First question is to "Estimate the Environmental Lapse Rate (ELR) by using linear regression (to 4 significant figures)." This is the rate temperature changes against altitude. I found the linear regression model (altitude~temperature) as $y=-205.2904699x+4803.389837$ and then found the ELR as $\frac{1}{-205.2904699}=-0.004871$ but their answer was $-0.004781$. As they're so similar, it looks like this is a typo.

Question 2: "Estimate the 0oC elevation (i.e. elevation where the mean air temperature is 0oC) using your locally determined ELR. Note: this estimation assumes that rises in air temperature are constant throughout the troposphere"

We answered this as 4803.39 but it accepted it and said the correct answer is 4836.36. Strange, I know, but this also happened to someone else on her course who had a difference of 50.

Question 3: "By how much has the 0oC elevation risen since the 1960s when the observed mean air temperature near the base of the Rwenzori Mountains (1095 mamsl) was 17.4oC rather than 18.1oC?"

I put $17.4$ and $18.1$ into the model and found the difference as $143.7033$, but the 'correct' answer is $146.4129$. Using their answer for question 1, I changed the model to set the coefficient as $\frac{1}{-0.004781}=-209.161$ and this did in fact give the 'correct' answer as $146.4129$.

Question 4: "The main icefields of the Rwenzori Mountains terminate below the peak elevation of approximately 4900 masml. By how much will surface water temperatures need to rise near the base of the mountains for the 0oC elevation to reach these last remaining icefields."

I solved $4900=-205.2904699x+4803.389837$ giving $x=-0.4706$ so the temperature would have to rise by 0.4706. Their 'correct' answer was $0.304$. I then repeated this with my adapted model (using the suspected typo in Q1 to find the 'correct' coefficient) and got an answer of $-0.4619$. This one has completely thrown me, did I do something wrong? I suspect I have as both my answer and the answer from the typo model are incorrect.

So, is this a typo?

Considering your data $$\left( \begin{array}{cc} x & y \\ 10.6 & 2565 \\ 11.2 & 2591 \\ 10.4 & 2595 \\ 6.9 & 3263 \\ 8.3 & 3411 \\ 1.2 & 4508 \\ 1.4 & 4468 \\ 5.1 & 3747 \\ 5.5 & 3556 \\ 7.0 & 3499 \\ 9.8 & 2931 \\ 14.9 & 1551 \\ 18.1 & 1095 \end{array} \right)$$
you performed the fit $$y=4803.39 -205.29 x$$ which results from the minimization of $$SSQ_1=\sum_i (a+bx_i-y_i)^2$$ while what they did is to regress $$x=23.1214 -0.00478073 y$$ which results from the minimization of $$SSQ_2=\sum_i (a+by_i-x_i)^2$$ which is totally different.