Question 11 OR - A-Level Maths

CAIE FP2 2009 June — Question 11 OR

Exam Board	CAIE
Module	FP2 (Further Pure Mathematics 2)
Year	2009
Session	June
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	T-tests (unknown variance)
Type	Two-sample t-test equal variance
Difficulty	Standard +0.8 This is a standard two-sample t-test question from Further Maths Statistics requiring hypothesis setup, assumptions, critical region calculation, and an inverse problem finding the maximum value of a. While methodical, it requires understanding of pooled variance, degrees of freedom, and working backwards from the test statistic—more demanding than typical A-level statistics but routine for Further Maths students who have practiced this topic.
Spec	5.05d Confidence intervals: using normal distribution

A study was made of the acidity levels in farmland on opposite sides of an island. The levels were measured at six randomly chosen points on the eastern side and at five randomly chosen points on the western side. The values obtained, in suitable units, are denoted by $x _ { E }$ and $x _ { W }$ respectively. The sample means $\bar { x } _ { E }$ and $\bar { x } _ { W }$, and unbiased estimates of the two population variances, $s _ { E } ^ { 2 }$ and $s _ { W } ^ { 2 }$, are as follows. $$\bar { x } _ { E } = 5.035 , s _ { E } ^ { 2 } = 0.0231 , \bar { x } _ { W } = 4.782 , s _ { W } ^ { 2 } = 0.0195 .$$ The population means on the eastern and western sides are denoted by $\mu _ { E }$ and $\mu _ { W }$ respectively. State suitable hypotheses for a test for a difference between the mean acidity levels on the two sides of the island. Stating any required assumptions, obtain the rejection region for a test at the $5 \%$ significance level of whether the mean acidity levels differ on the two sides of the island. Give the conclusion of the test. Find the largest value of $a$ for which the samples above provide evidence at the $5 \%$ significance level that $\mu _ { E } - \mu _ { W } > a$.

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Question 11 (EITHER - Mechanics, 14 marks):

Part 1 (3 marks)

Answer	Marks	Guidance
Answer	Mark	Guidance
$T = 4mgx/a$	B1	Find tension $T$
$m\,\mathrm{d}^2x/\mathrm{d}t^2 = mg - T$	M1	Apply Newton's law of motion to $B$
$\mathrm{d}^2x/\mathrm{d}t^2 = -(g/a)(4x-a)$	A1	Combine. A.G.

Part 2 (3 marks)

Answer	Marks	Guidance
Answer	Mark	Guidance
$\mathrm{d}^2y/\mathrm{d}t^2 = -(4g/a)\,y$	M1 A1	Substitute e.g. $y = x - \tfrac{1}{4}a$ and rearrange
$x_c = \tfrac{1}{4}a$	A1	State centre of motion, or derive from $y=0$

Part 3 (8 marks)

Answer	Marks	Guidance
Answer	Mark	Guidance
$T = \tfrac{1}{3}mg,\; x_s = a/12$	M1 A1	Find $x$ when $A$ starts to slip using $F = \mu R$
$y = y_{max}\cos\omega t$ or $y_{max}\sin\omega t$	M1	Valid use of SHM equation to find $t_s$ to slipping
$y_s = y_{max}\cos\omega t_s$	M1	EITHER: Valid use of cosine form
$t_s = t_1 - t_2,\; y_{max} = y_{max}\sin\omega t_1$	M1	OR: Valid use of sine form
$y_s = y_{max}\sin\omega t_2$	(M1)
$y_s/y_{max} = (x_c - x_s)/x_c$	M1	Substitute for $y_s$, $y_{max}$
$= (a/6)/(a/4) = \tfrac{2}{3}$	A1
$t_s = (\cos^{-1}\tfrac{2}{3})/\omega$
or $(\tfrac{1}{2}\pi - \sin^{-1}\tfrac{2}{3})/\omega$	M1
$t_s = 0.421\sqrt{a/g}$	A1	Substitute $\omega = 2\sqrt{g/a}$ and evaluate

Question 11 (OR - Statistics, 14 marks):

Part 1 (9 marks)

Answer	Marks	Guidance
Answer	Mark	Guidance
$H_0: \mu_E = \mu_W,\; H_1: \mu_E \neq \mu_W$	B1	State hypotheses
Two populations have Normal distributions and common variance	B1	State assumption [A.E.F.]
$\sigma^2 = (5\times0.0231 + 4\times0.0195)/9$	M1	Estimate common variance
$= 0.0215$ or $0.1473^2$	A1
$t_{9,\,0.975} = 2.26[2]$	B1	Use correct tabular value of $t$
$\lvert\bar{X}_E - \bar{X}_W\rvert \geq t\sigma\sqrt{1/6+1/5}$	M1	Formulate rejection region (with any $t$; allow $>$)
$= 0.201$	A1
$0.253 > 0.201\sqrt{\phantom{x}}$		Compare actual sample means with region
or $2.85 > 2.26[2]\sqrt{\phantom{x}}$	M1 A1	or compare calculated $t$ with tabular $t$
Mean acidity levels do differ	A1	Consistent conclusion (A.E.F.; dep values above)

Part 2 (4 marks)

Answer	Marks	Guidance
Answer	Mark	Guidance
$\bar{X}_E - \bar{X}_W - a \geq t\sigma\sqrt{1/6+1/5}$	M1	State condition on $a$ (with any $t$; allow $>$ or $=$)
$t_{9,\,0.95} = 1.83[3]$	A1	Use correct tabular value of $t$
$a_{max} = 0.253 - 0.163 = 0.09$ (2 dp)	M1 A1	Substitute to find largest value of $a$

# Question 11 (EITHER - Mechanics, 14 marks):

## Part 1 (3 marks)

| Answer | Mark | Guidance |
|--------|------|----------|
| $T = 4mgx/a$ | B1 | Find tension $T$ |
| $m\,\mathrm{d}^2x/\mathrm{d}t^2 = mg - T$ | M1 | Apply Newton's law of motion to $B$ |
| $\mathrm{d}^2x/\mathrm{d}t^2 = -(g/a)(4x-a)$ | A1 | Combine. **A.G.** |

## Part 2 (3 marks)

| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathrm{d}^2y/\mathrm{d}t^2 = -(4g/a)\,y$ | M1 A1 | Substitute e.g. $y = x - \tfrac{1}{4}a$ and rearrange |
| $x_c = \tfrac{1}{4}a$ | A1 | State centre of motion, or derive from $y=0$ |

## Part 3 (8 marks)

| Answer | Mark | Guidance |
|--------|------|----------|
| $T = \tfrac{1}{3}mg,\; x_s = a/12$ | M1 A1 | Find $x$ when $A$ starts to slip using $F = \mu R$ |
| $y = y_{max}\cos\omega t$ *or* $y_{max}\sin\omega t$ | M1 | Valid use of SHM equation to find $t_s$ to slipping |
| $y_s = y_{max}\cos\omega t_s$ | M1 | EITHER: Valid use of cosine form |
| $t_s = t_1 - t_2,\; y_{max} = y_{max}\sin\omega t_1$ | M1 | OR: Valid use of sine form |
| $y_s = y_{max}\sin\omega t_2$ | (M1) | |
| $y_s/y_{max} = (x_c - x_s)/x_c$ | M1 | Substitute for $y_s$, $y_{max}$ |
| $= (a/6)/(a/4) = \tfrac{2}{3}$ | A1 | |
| $t_s = (\cos^{-1}\tfrac{2}{3})/\omega$ | | |
| *or* $(\tfrac{1}{2}\pi - \sin^{-1}\tfrac{2}{3})/\omega$ | M1 | |
| $t_s = 0.421\sqrt{a/g}$ | A1 | Substitute $\omega = 2\sqrt{g/a}$ and evaluate |

---

# Question 11 (OR - Statistics, 14 marks):

## Part 1 (9 marks)

| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu_E = \mu_W,\; H_1: \mu_E \neq \mu_W$ | B1 | State hypotheses |
| Two populations have Normal distributions and common variance | B1 | State assumption [A.E.F.] |
| $\sigma^2 = (5\times0.0231 + 4\times0.0195)/9$ | M1 | Estimate common variance |
| $= 0.0215$ *or* $0.1473^2$ | A1 | |
| $t_{9,\,0.975} = 2.26[2]$ | B1 | Use correct tabular value of $t$ |
| $\lvert\bar{X}_E - \bar{X}_W\rvert \geq t\sigma\sqrt{1/6+1/5}$ | M1 | Formulate rejection region (with any $t$; allow $>$) |
| $= 0.201$ | A1 | |
| $0.253 > 0.201\sqrt{\phantom{x}}$ | | Compare actual sample means with region |
| *or* $2.85 > 2.26[2]\sqrt{\phantom{x}}$ | M1 A1 | *or* compare calculated $t$ with tabular $t$ |
| Mean acidity levels **do** differ | A1 | Consistent conclusion (A.E.F.; dep values above) |

## Part 2 (4 marks)

| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{X}_E - \bar{X}_W - a \geq t\sigma\sqrt{1/6+1/5}$ | M1 | State condition on $a$ (with any $t$; allow $>$ or $=$) |
| $t_{9,\,0.95} = 1.83[3]$ | A1 | Use correct tabular value of $t$ |
| $a_{max} = 0.253 - 0.163 = 0.09$ (2 dp) | M1 A1 | Substitute to find largest value of $a$ |

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A study was made of the acidity levels in farmland on opposite sides of an island. The levels were measured at six randomly chosen points on the eastern side and at five randomly chosen points on the western side. The values obtained, in suitable units, are denoted by $x _ { E }$ and $x _ { W }$ respectively. The sample means $\bar { x } _ { E }$ and $\bar { x } _ { W }$, and unbiased estimates of the two population variances, $s _ { E } ^ { 2 }$ and $s _ { W } ^ { 2 }$, are as follows.

$$\bar { x } _ { E } = 5.035 , s _ { E } ^ { 2 } = 0.0231 , \bar { x } _ { W } = 4.782 , s _ { W } ^ { 2 } = 0.0195 .$$

The population means on the eastern and western sides are denoted by $\mu _ { E }$ and $\mu _ { W }$ respectively. State suitable hypotheses for a test for a difference between the mean acidity levels on the two sides of the island.

Stating any required assumptions, obtain the rejection region for a test at the $5 \%$ significance level of whether the mean acidity levels differ on the two sides of the island. Give the conclusion of the test.

Find the largest value of $a$ for which the samples above provide evidence at the $5 \%$ significance level that $\mu _ { E } - \mu _ { W } > a$.

\hfill \mbox{\textit{CAIE FP2 2009 Q11 OR}}

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Q1 5 Q2 7 Q3 8 Q4 11 Q5 12 Q6 6 Q7 8 Q8 8 Q9 9 Q10 12 Q11 EITHER Q11 OR