# Question 5:

## Part (a):
Calorific value depends upon moisture content; Moisture (content) is set/are fixed values | B1 | 1 | Must be in context; **not** "it", etc. Use of $x$ and $y \Rightarrow$ B0

## Part (b):
$b \text{ (gradient)} = -0.076$ | B2 (B1) | | AWRT; including $-$ve sign $(-0.07582)$; AWFW; including $-$ve sign; Treat rounding of correct answers as ISW

$b \text{ (gradient)} = -0.07$ to $-0.08$

$a \text{ (intercept)} = 5.35$ to $5.36$ | B2 (B1) | | AWFW (5.35385); AWFW

$a \text{ (intercept)} = 5.1$ to $5.6$

Thus $y = (5.35 \text{ to } 5.36) - 0.076x$ | BF1 | 5 | F on $a$ and $b$ even if rounded

**Alternative solutions:**

Attempt at $\sum x$, $\sum x^2$, $\sum y$ & $\sum xy$ $(\sum y^2)$ or Attempt at $S_{xx}$ & $S_{xy}$ $(S_{yy})$ | M1 | | 455, 20475, 35.1 & 883.5 (121.33) (all 4 attempted); 4550 & $-345$ (26.56) (both attempted)

Attempt at correct formula for $b$ (gradient); $b \text{ (gradient)} = -0.076$; $a \text{ (intercept)} = 5.35$ to $5.36$; Thus $y = (5.35 \text{ to } 5.36) - 0.076x$ | m1, A1, A1, BF1 | 5 | AWRT; AWFW; F on $a$ and $b$ even if rounded

Notes: **1** If $a$ and $b$ interchanged and equation $y = ax + b$ used $\Rightarrow$ max of 5 marks; **2** If $a$ and $b$ interchanged and equation $y = a + bx$ used $\Rightarrow$ maximum of BF1; **3** Marks lost here cannot be gained from subsequent work in parts (d) and/or (e)

## Part (c):
$a$: calorific value of wood with zero/no moisture or dry maximum calorific value | B1 | | OE; $a \leq 0 \Rightarrow$ B0

$b$: each $1(\%)$ rise in moisture content reduces calorific value by $0.076$ MWh/tonne | B2 | 3 | In context and with values; F on $b$; $b \geq 0 \Rightarrow$ B0

As $x$ increases $y$ decreases | (B1) | | Negative relationship/correlation

## Part (d):
$y_{27} = 3.28$ to $3.32$ | B2 | 2 | AWFW (3.30659)

$= 2.5$ to $3.5$ | (B1) | | AWFW; even if by interpolation from original data giving likely values of 3 or 3.04

**Alternative:**
$y_{27} = (5.35 \text{ to } 5.36) - 0.076 \times 27$ | M1 | | Clear evidence of correct use of c's equation with $x = 27$

$= 3.28$ to $3.32$ | A1 | 2 | AWFW (3.30659)

## Part (e):
$r(35, 2.5) = -0.21$ to $-0.19$ | B2 | 2 | AWFW; including $-$ve sign $(-0.20000)$

$= 0.1$ to $0.3$ | (B1) | | AWFW; ignore sign

**Alternative:**
$r(35, 2.5) = 2.5 - y_{35}$ | M1 | | Used; allow $y_{35} - 2.5$

$= 2.5 - \{(5.35 \text{ to } 5.36) - 0.076 \times 35\}$ | | |

$= -0.21$ to $-0.19$ | A1 | 2 | AWFW $(-0.20000)$

## Part (f):
Good/reasonable/accurate/correct/etc; Accept more positive qualifying adjectives | B1 | 1 | OE; ignore reasoning; Very good (B1); Not good (B0)

## Part (g)(i):
Extrapolation/outside (observed) range (of $x$) | B1 | 1 | OE

## Part (g)(ii):
$y_{80} = -0.5$ to $-1$ | B1 | | AWFW $(-0.71209)$

Negative value for calorific value is impossible OR More energy needed than is generated | Bdep1 | 2 | OE; dependent on B1; Must be in context; negative value impossible $\Rightarrow$ Bdep0

**Total: 17 marks**

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