## Question 6:

### Part (a):
$H_0: \mu_{\text{new}} - \mu_{\text{old}} = 8$ | B1 | Accept equivalent rearranged equation; definitions of parameters must be clear e.g. use of 1 and 2 without definitions scores B0; accept '$x$' for 'new' and '$y$' for 'old' as defined in the question

$H_1: \mu_{\text{new}} - \mu_{\text{old}} > 8$ | B1 | Accept equivalent rearranged strict inequality; definitions of parameters must be clear

$z = \frac{\pm(83 - 74 - 8)}{\sqrt{\frac{7}{50} + \frac{6}{40}}}$ | M1 | M1 for attempting standard error; condone swapping 7 and 6; accept 6.86 and 5.85 for 7 and 6

$z = \frac{\pm 1}{0.5385\ldots} = 1.86$ | dM1A1 | dM1 for $1/\text{"their standard error"}$; A1 for awrt 1.86; NB $-1.86$ is A0; if 8 missing from $H_1$ then accept $z = \frac{9}{0.5385\ldots} =$ awrt 16.7 and must be consistent with their $H_1$

cv $z = 1.6449$ | B1 | Accept $\pm$ or probability of 0.9686

$(1.86 > 1.6449)$ so reject $H_0$ | | (Or $0.95 < 0.9686$)

Evidence to support engineer's claim (that the new battery will last more than 8 hours longer than the old battery) | A1cso | Correct comment in context; must mention "engineers claim" or "battery", "old", "new" and "8"

### Part (b):
Sample sizes are large | B1 |

CLT guarantees sample means ($\bar{X}$ and $\bar{Y}$) are approximately normally distributed | B1 | Must mention **means** and **normal**; no assumptions are being made so B0 if key to answer