| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | $[\bar{y} =] \frac{-27}{12} = -2.25$, $\text{Var}(Y) = \frac{62.98}{12} - (-2.25)^2 = 0.1858333\ldots$ (allow $\frac{223}{1200}$) $\underline{\text{awrt } 0.186}$ | B1, M1, A1 | B1 either fraction or exact decimal equivalent [must see mean separately to earn this mark]. M1 for expr' for variance $\frac{62.98}{12} - \bar{y}^2$ [ft $\bar{y}$] or $\frac{S_{yy}}{11} = \text{awrt } 0.203$) [No $\sqrt{}$]. For M1 in (b)(i) and 1st M1 in (c) do not allow ft for $S_{yy} = -1190.7$ |
| | | (3) | |
| (b)(i) | $S_{xy} = -1190.7 - \frac{(504)(-27)}{12}$ or $-56.7$ | B1 | Correct expression for $S_{xy}$ or $-56.7$ (May be implied by a correct value for $r$) |
| | $r = \frac{\text{'}-56.7'}{\sqrt{(1674)(2.23)}} = \text{, } -0.9280105\ldots$ | M1, A1 | M1 for correct express' for $r$ with 1674, 2.23 and their $S_y$. [Correct ans. only 3/3, $r = -0.93$ is 2/3] |
| | | $\underline{\text{awrt } -0.928}$ | |
| (ii) | Negative correlation, so Priya's belief is incorrect. | B1 | B1 for Priya's belief not supported and reason e.g. negative correlation or $r$ is negative or $r$ is close to $-1$ or as salinity (or $x$) increases, temperature (or $y$) decreases |
| | | (4) | |
| (c) | $b = \frac{\text{'}-56.7'}{1674}$ [$= -0.033870\ldots$] | M1 | |
| | $\frac{-27}{12} = a + b' \times \frac{504}{12}$ or $a = -2.25 - \text{'}-0.03387...' \times 42$, $a = \text{awrt } -0.827$ | M1, A1 | 1st M1 for correct expression for $b$ f.t. their $S_{yy}$ (May be implied by correct answer). 2nd M1 for correct use of $a = \bar{y} - b\bar{x}$ to find $a$ (f.t. their value of $b$)(Implied by $-0.827$). 1st A1 for $a = \text{awrt } -0.827$ (no fraction). 2nd A1 for an equ'n in the form $y = a + bx$ with their $a$ and $b = \text{awrt } -0.0339$ (no fraction) |
| | $y = -0.827 - 0.0339x$ | A1 | |
| | | (4) | |
| (d) | $[y = -0.827 - 0.0339(32) =] -1.9°C$ | B1 | awrt $-1.9$ (no fractions) |
| | | (1) | |
| (e) | $\frac{(w-32)}{1.8} = -0.827 - 0.339x$ (o.e.) | M1 | M1 for substituting $\frac{(w-32)}{1.8}$ for $y$ (o.e.) in their regression equation |
| | | | |
| | $w = 30.5 - 0.061x$ | A1 | A1 for a correct equation for $w$ in terms of $x$ with $c = \text{awrt } 31$ and $d = \text{awrt } -0.061$ |
| | | (2) | |
| (f)(i) | $\text{Var}(W) = 1.8^2 \text{ Var}(Y)$, $= 0.602\ldots$ | M1, A1 | M1 for $1.8^2 \times \text{Var}(Y)$ f.t. their "(a)" (if $> 0$) [[Allow use of $s^2 = \text{awrt } 0.66$ to score M1A1] |
| (ii) | $r_{yx} = r_{wx} = -0.928$ | B1ft | (3) |
| | | [17 marks] | |