| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | $[\text{Let } F \sim N(166.5, 6.1^2)]$ $P(F < k) = 0.01 \Rightarrow \frac{k - 166.5}{6.1} = -2.3263$ | M1 | M1 for standardising (allow $\pm$) with $k, 166.5$ and $6.1$ and set equal to a $z$ value $2.3 < \|z\| < 2.4$ |
| | $k = 152.309...$ $\mathbf{152}$ or awrt $\mathbf{152.3}$ | A1 | A1 for 152 or awrt 152.3 **Ans only 2/2** [Condone poor use of notation e.g. $P(\frac{k-166.5}{6.1}) = -2.3263$] **Allow percentages instead of probabilities throughout.** |
| (b) | $[P(150 < F < 175) =] 0.914840...$ | B1 | B1 for awrt 0.915 |
| | awrt $\mathbf{0.915}$ | | |
| (c) | $P(F > 160 \mid 150 < F < 175)$ | M1 | 1st M1 for interpreting demand as an appropriate conditional probability ($\Rightarrow$ by 2nd M1) |
| | $= \frac{P(160 < F < 175)}{P(150 < F < 175)}$ or $\frac{P(160 < F < 175)}{"{b}"}$ | M1 | 2nd M1 for correct ratio of expressions (can ft their (b) on denominator) ($\Rightarrow$ by 1st A1ft) |
| | $= \frac{0.7749487...}{"0.91484..."}$ | A1ft | 1st A1ft for a correct ratio of probs (can ft their "0.9148..." to 3sf from (b) if $> 0.775$) |
| | $= 0.84708...$ awrt $\mathbf{0.847}$ | A1 | 2nd A1 for awrt 0.847 |
| (d) | $H_0 : \mu = 166.5$ $H_1 : \mu < 166.5$ | B1 | B1 for both correct hypotheses in terms of $\mu$ |
| | $[\text{Let } X = \text{height of female from 2nd country}] \bar{X} \sim N\left(166.5, \left(\frac{7.4}{\sqrt{50}}\right)^2\right)$ | M1 | 1st M1 for selecting the correct model (needn't use $\bar{X}$ $\Rightarrow$ by standardisation or 1st A1) |
| | $P(\bar{X} < 164.6) = 0.03472...$ | A1 | 1st A1 for correct use of the correct model i.e. awrt 0.035 (allow 0.04 if $P("\bar{X}" < 164.6)$seen). Condone $P("\bar{X}" > 164.6) = 0.9652$ or awrt 0.97 **only if comparison with 0.95 is made** |
| | $[0.0347... < 0.05$ so significant or reject $H_0$] | dA1 | dA1 (dep on M1A1 only) for a correct inference in context. **Must mention Mia's belief or mean height of females/women.** **Do NOT award if contradictory statements about hypotheses made** e.g. "not sig" |
| | There is evidence to support Mia's belief | | |
| | *Notes* | | |
| | M0 for $\bar{X} \sim N(164.6, \ldots)$ If they achieve $p = $ awrt 0.035 (o.e. with z-value or CV of 166.3) and a correct conclusion in context is given score **M0A0A1** [and SC for awrt 0.97 $> 0.95$ case] | | |
| | **ALT:** Use of $z$ value: Need to see $Z = -1.8(15\ldots)$ and cv of $1.6449$ (allow 1.64 or better) for 1st A1. **ALT:** Use of CR or CV for $\bar{X}$: Need to see "$\bar{X}$" $< 164.7786\ldots$ or CV $= \ldots$ (awrt 164.8) for 1st A1. Condone truncation i.e. 164.7 or better. 2nd dA1 (dep on M1A1 only) for a correct inference in context. Must mention Mia's belief or mean height of females/women. Do NOT award if contradictory statements about hypotheses made e.g. "not sig". SC: **M0 for $\bar{X} \sim N(164.6, \ldots)$** If they achieve $p = $ awrt 0.035 (o.e. with z-value or CV of 166.3) and a correct conclusion in context is given score **M0A0A1** [and SC for awrt 0.97 $> 0.95$ case] |