CAIE FP2 2011 June — Question 9 9 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2011
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicT-tests (unknown variance)
TypeTwo-sample z-test large samples
DifficultyChallenging +1.2 This is a standard two-sample t-test with summary statistics requiring calculation of sample means and variances, pooled variance, test statistic, and comparison with critical value. While it involves multiple computational steps and understanding of hypothesis testing framework, it follows a routine procedure taught in Further Statistics with no novel insight required. The 2.5% significance level and clear one-tailed context make it slightly more straightforward than average A-level questions.
Spec5.05c Hypothesis test: normal distribution for population mean
9 Mr Lee asserts that boys are slower than girls at completing a particular mathematical puzzle. In order to test his assertion, a random sample of 40 boys and a random sample of 60 girls are selected from a large group of students who attempted the puzzle. The times taken by the boys, \(b\) minutes, and the times taken by the girls, \(g\) minutes, are summarised as follows. $$\Sigma b = 92.0 \quad \Sigma b ^ { 2 } = 216.5 \quad \Sigma g = 129.8 \quad \Sigma g ^ { 2 } = 288.8$$ Test at the \(2.5 \%\) significance level whether this evidence supports Mr Lee's assertion.
Question 9:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(H_0: \mu_b = \mu_g\), \(H_1: \mu_b > \mu_g\)B1 State hypotheses
\(s_b^2 = (216.5 - 92.0^2/40)/39\ [= 0.126\) or \(0.354^2]\)M1 Estimate population variance using boys' sample (allow biased: \(\sigma_{b,40}^2 = 0.1225\) or \(0.35^2\))
\(s_g^2 = (288.8 - 129.8^2/60)/59\ [= 0.136\) or \(0.368^2]\)M1 Estimate population variance using girls' sample (allow biased: \(\sigma_{g,60}^2 = 0.133\) or \(0.365^2\))
\(s^2 = s_b^2/40 + s_g^2/60 = 0.00541\) or \(0.0736^2\)M1 A1 Estimate population variance for combined sample
\(z = (2.3 - 2.163)/s = 0.1367/0.0736 = 1.86\) (or \(1.88\))M1 A1 Calculate value of \(z\) (to 2 dp)
S.R. Allow implicit assumption of equal variances:
Find pooled estimate: \(s^2: (40\sigma_{b,40}^2 + 60\sigma_{g,60}^2)/98 = 0.132\)(M1A1)
\(z = (2.3 - 2.163)/s\sqrt{(1/40+1/60)} = 1.85\)(M1 A1)
\(z_{0.975} = 1.96\)B1 Compare with correct tabular \(t\) value
Claim is not correctA1\(\sqrt{}\) Conclusion consistent with values (A.E.F.)
Total: [9]
## Question 9:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \mu_b = \mu_g$, $H_1: \mu_b > \mu_g$ | B1 | State hypotheses |
| $s_b^2 = (216.5 - 92.0^2/40)/39\ [= 0.126$ or $0.354^2]$ | M1 | Estimate population variance using boys' sample (allow biased: $\sigma_{b,40}^2 = 0.1225$ or $0.35^2$) |
| $s_g^2 = (288.8 - 129.8^2/60)/59\ [= 0.136$ or $0.368^2]$ | M1 | Estimate population variance using girls' sample (allow biased: $\sigma_{g,60}^2 = 0.133$ or $0.365^2$) |
| $s^2 = s_b^2/40 + s_g^2/60 = 0.00541$ or $0.0736^2$ | M1 A1 | Estimate population variance for combined sample |
| $z = (2.3 - 2.163)/s = 0.1367/0.0736 = 1.86$ (or $1.88$) | M1 A1 | Calculate value of $z$ (to 2 dp) |
| **S.R.** Allow implicit assumption of equal variances: | | |
| Find pooled estimate: $s^2: (40\sigma_{b,40}^2 + 60\sigma_{g,60}^2)/98 = 0.132$ | (M1A1) | |
| $z = (2.3 - 2.163)/s\sqrt{(1/40+1/60)} = 1.85$ | (M1 A1) | |
| $z_{0.975} = 1.96$ | B1 | Compare with correct tabular $t$ value |
| Claim is not correct | A1$\sqrt{}$ | Conclusion consistent with values (A.E.F.) |

**Total: [9]**

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9 Mr Lee asserts that boys are slower than girls at completing a particular mathematical puzzle. In order to test his assertion, a random sample of 40 boys and a random sample of 60 girls are selected from a large group of students who attempted the puzzle. The times taken by the boys, $b$ minutes, and the times taken by the girls, $g$ minutes, are summarised as follows.

$$\Sigma b = 92.0 \quad \Sigma b ^ { 2 } = 216.5 \quad \Sigma g = 129.8 \quad \Sigma g ^ { 2 } = 288.8$$

Test at the $2.5 \%$ significance level whether this evidence supports Mr Lee's assertion.

\hfill \mbox{\textit{CAIE FP2 2011 Q9 [9]}}
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