S1 January 2013 Q3
3. A biologist is comparing the intervals (\(m\) seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (\(t\) °C). The following results were obtained.
| \(t\) °C | 8 | 13 | 14 | 15 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|---|---|
| \(m\) secs | 6.5 | 4.5 | 6 | 5 | 4 | 3 | 2 | 1 |
(You may use \(\sum tm = 469.5\), \(\mathrm{S}_{tt} = 354\), \(\mathrm{S}_{mm} = 25.5\))
| Scheme | Marks |
|---|---|
| \(\sum t = 140\) (or \(\bar{t} = 17.5\)) and \(\sum m = 32\) (or \(\bar{m} = 4\)) | B1 B1 |
| \((\mathrm{S}_{tm}) = 469.5 - \dfrac{\text{"}140\text{"}\times\text{"}32\text{"}}{8}\) | M1 |
| \((\mathrm{S}_{tm} =) -90.5\) | A1cso |
| (4) |
Notes
1st B1 for 140 seen in correct context or correctly labelled
2nd B1 for 32 seen in correct context or correctly labelled.
(allow a fully correct expression – not “…+…+…”) 4480 used correctly is B1B1
M1 for attempting a correct expression. Follow through their 140 and their 32
You may see attempt at \(\sum(t - \bar{t})(m - \bar{m})\). This must have all the products seen.
A1cso requires a correct expression seen and no incorrect working leading to \(-90.5\)
| Scheme | Marks |
|---|---|
| \(b = \dfrac{\mathrm{S}_{tm}}{\mathrm{S}_{tt}} = \dfrac{-90.5}{354}\) | M1 |
| \(b = -0.255649\ldots\) (allow \(\frac{181}{708}\)) \(-0.25\) or awrt \(-0.26\) | A1 |
| \(a = \dfrac{\text{"}32\text{"}}{8} - b\times\dfrac{\text{"}140\text{"}}{8}\) | M1 |
| So equation of the line is \(\boldsymbol{m = 8.47 - 0.256t}\) (allow \(m = \frac{11999}{1416} - \frac{181}{708}t\)) | A1 |
| (4) |
Notes
1st M1 for a correct expression for \(b\). Follow through their \(\mathrm{S}_{tm}\). Condone missing “−”
1st A1 for awrt \(-0.26\) or condone \(-0.25\)
2nd M1 for a correct method for \(a\). Follow through their sums from part (a) and their value of \(b\)
2nd A1 for a correct equation for \(m\) and \(t\) with \(a\) = awrt 8.47 and \(b\) = awrt \(-0.256\). Must be an equation in \(m\) and \(t\), use of \(x\) or \(y\) scores A0 here.
| Scheme | Marks |
|---|---|
| \((8.47 - 0.256\times 10 =)\ 5.9\ldots\) awrt 5.9 | B1 |
| (1) |
Notes
B1 for awrt 5.9 Accept 6 if the correct expression (awrt \(8.47 - 10\times\) awrt 0.256) is seen
| Scheme | Marks |
|---|---|
| Should be reliable since 10 is in the range (of the data) | B1 |
| (1) | |
| (10 marks) |
Notes
B1 for suggesting it is reliable and mentioning 10 within the range (of the data.)
or suggesting it is reliable since interpolating or not extrapolating
NB “it is reliable since it is in the range” is B0 since “it” is not explicit enough
Condone extra non-relevant comments but penalise contradictory comments.
e.g. “near the extreme so not reliable but not extrapolated so reliable” is B0 since contradicts
“reliable since 10 is within the range (of temps) and 5.9 within range of times” is B1 since irrelevant