CAIE FP2 2012 November — Question 9 10 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2012
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicT-tests (unknown variance)
TypeTwo-sample z-test large samples
DifficultyStandard +0.3 This is a standard two-sample t-test with summary statistics provided. Students must calculate sample means and variances from the summations, apply the pooled variance formula, and conduct a hypothesis test—all routine A-level Further Statistics procedures requiring no novel insight, though slightly above average difficulty due to the computational steps involved.
Spec5.05c Hypothesis test: normal distribution for population mean
9 Experiments are conducted to test the breaking strength of each of two types of rope, \(P\) and \(Q\). A random sample of 50 ropes of type \(P\) and a random sample of 70 ropes of type \(Q\) are selected. The breaking strengths, \(p\) and \(q\), measured in appropriate units, are summarised as follows. $$\Sigma p = 321.2 \quad \Sigma p ^ { 2 } = 2120.0 \quad \Sigma q = 475.3 \quad \Sigma q ^ { 2 } = 3310.0$$ Test, at the \(10 \%\) significance level, whether the mean breaking strengths of type \(P\) and type \(Q\) ropes are the same.
Question 9:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(H_0: \mu_P = \mu_Q\), \(H_1: \mu_P \neq \mu_Q\)B1 State hypotheses
\(s_P^2 = (2120 - 321 \cdot 2^2/50)/49\)M1 Estimate population variance using P's sample
\(= 1 \cdot 132\) or \(1 \cdot 064^2\) \([= 1 \cdot 155\) or \(1 \cdot 075^2]\) Allow use of biased
\(s_Q^2 = (3310 - 475 \cdot 3^2/70)/69\)M1 Estimate population variance using Q's sample
\(= 1 \cdot 182\) or \(1 \cdot 087^2\) \([= 1 \cdot 199\) or \(1 \cdot 095^2]\) Allow use of biased
\(s^2 = s_P^2/50 + s_Q^2/70 = 0 \cdot 04023\) or \(0 \cdot 2006^2\)M1 A1 Allow use of \(\sigma_{P,50}^2\), \(\sigma_{Q,70}^2\) giving \(0 \cdot 03949\) or \(0 \cdot 1987^2\)
\(z = (6 \cdot 424 - 6 \cdot 79)/s = -0 \cdot 366/0 \cdot 2006 = -1 \cdot 82[5]\)M1 A1 Calculate value of \(z\) to 2 d.p., either sign
\((or -1 \cdot 84)\)A1
S.R. Pooled: \(s^2 = (50\sigma_{P,50}^2 + 70\sigma_{Q,70}^2)/118 = 1 \cdot 180\) or \(1 \cdot 086^2\)(M1A1) Equal variances assumption — allow implicitly, deduct A1 if not explicit
\(z = (6 \cdot 424 - 6 \cdot 79)/s\sqrt{(1/50 + 1/70)} = -1 \cdot 82\)(M1 A1)(A1)
\(z_{0.95} = 1 \cdot 645\) (to 2 d.p.)B1 State or use correct tabular \(z\) value
Breaking strengths not the sameA1✓ Conclusion consistent with values (A.E.F.) 
# Question 9:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu_P = \mu_Q$, $H_1: \mu_P \neq \mu_Q$ | B1 | State hypotheses |
| $s_P^2 = (2120 - 321 \cdot 2^2/50)/49$ | M1 | Estimate population variance using P's sample |
| $= 1 \cdot 132$ or $1 \cdot 064^2$ $[= 1 \cdot 155$ or $1 \cdot 075^2]$ | | Allow use of biased |
| $s_Q^2 = (3310 - 475 \cdot 3^2/70)/69$ | M1 | Estimate population variance using Q's sample |
| $= 1 \cdot 182$ or $1 \cdot 087^2$ $[= 1 \cdot 199$ or $1 \cdot 095^2]$ | | Allow use of biased |
| $s^2 = s_P^2/50 + s_Q^2/70 = 0 \cdot 04023$ or $0 \cdot 2006^2$ | M1 A1 | Allow use of $\sigma_{P,50}^2$, $\sigma_{Q,70}^2$ giving $0 \cdot 03949$ or $0 \cdot 1987^2$ |
| $z = (6 \cdot 424 - 6 \cdot 79)/s = -0 \cdot 366/0 \cdot 2006 = -1 \cdot 82[5]$ | M1 A1 | Calculate value of $z$ to 2 d.p., either sign |
| $(or -1 \cdot 84)$ | A1 | |
| **S.R.** Pooled: $s^2 = (50\sigma_{P,50}^2 + 70\sigma_{Q,70}^2)/118 = 1 \cdot 180$ or $1 \cdot 086^2$ | (M1A1) | Equal variances assumption — allow implicitly, deduct A1 if not explicit |
| $z = (6 \cdot 424 - 6 \cdot 79)/s\sqrt{(1/50 + 1/70)} = -1 \cdot 82$ | (M1 A1)(A1) | |
| $z_{0.95} = 1 \cdot 645$ (to 2 d.p.) | B1 | State or use correct tabular $z$ value |
| Breaking strengths not the same | A1✓ | Conclusion consistent with values (A.E.F.) |

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9 Experiments are conducted to test the breaking strength of each of two types of rope, $P$ and $Q$. A random sample of 50 ropes of type $P$ and a random sample of 70 ropes of type $Q$ are selected. The breaking strengths, $p$ and $q$, measured in appropriate units, are summarised as follows.

$$\Sigma p = 321.2 \quad \Sigma p ^ { 2 } = 2120.0 \quad \Sigma q = 475.3 \quad \Sigma q ^ { 2 } = 3310.0$$

Test, at the $10 \%$ significance level, whether the mean breaking strengths of type $P$ and type $Q$ ropes are the same.

\hfill \mbox{\textit{CAIE FP2 2012 Q9 [10]}}
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