CAIE FP2 2011 June — Question 7 8 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2011
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicT-tests (unknown variance)
TypeSingle sample t-test
DifficultyStandard +0.3 This is a straightforward one-sample t-test with clear hypotheses (H₀: μ = 1.2 vs H₁: μ > 1.2), requiring calculation of sample mean and standard deviation from given data, then applying the standard t-test procedure at a stated significance level. While it involves multiple computational steps and understanding of hypothesis testing framework, it follows a completely standard template with no conceptual subtleties or novel problem-solving required, making it slightly easier than average.
Spec5.05c Hypothesis test: normal distribution for population mean

7 A greengrocer claims that his cabbages have a mean mass of more than 1.2 kg . In order to check his claim, he weighs 10 cabbages, chosen at random from his stock. The masses, in kg, are as follows. $$\begin{array} { l l l l l l l l l l } 1.26 & 1.24 & 1.17 & 1.23 & 1.18 & 1.25 & 1.19 & 1.20 & 1.21 & 1.18 \end{array}$$ Stating any assumption that you make, test at the \(10 \%\) significance level whether the greengrocer's claim is supported by this evidence.

Question 7:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Population of masses has normal distributionB1 State valid assumption (A.E.F.)
\(H_0: \mu = 1.2\), \(H_1: \mu > 1.2\)B1 State both hypotheses
\(\bar{x} = 12.11/10\ [= 1.211]\)B1 Calculate sample mean
\(s^2 = (14.6745 - 12.11^2/10)/9\)M1 Estimate population variance
\([= 0.00103\) or \(0.0321^2]\) (allow biased: \(0.000929\) or \(0.0305^2\))
\(t = (\bar{x} - 1.2)/(s/\sqrt{10}) = 1.08\)M1 A1 Calculate value of \(t\) (to 2 dp)
\(t_{9,\,0.90} = 1.38\ [3]\)B1 Compare with correct tabular \(t\) value
Claim is not correctA1\(\sqrt{}\) Conclusion consistent with values (A.E.F.)
Total: [8]
## Question 7:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Population of masses has normal distribution | B1 | State valid assumption (A.E.F.) |
| $H_0: \mu = 1.2$, $H_1: \mu > 1.2$ | B1 | State both hypotheses |
| $\bar{x} = 12.11/10\ [= 1.211]$ | B1 | Calculate sample mean |
| $s^2 = (14.6745 - 12.11^2/10)/9$ | M1 | Estimate population variance |
| $[= 0.00103$ or $0.0321^2]$ | | (allow biased: $0.000929$ or $0.0305^2$) |
| $t = (\bar{x} - 1.2)/(s/\sqrt{10}) = 1.08$ | M1 A1 | Calculate value of $t$ (to 2 dp) |
| $t_{9,\,0.90} = 1.38\ [3]$ | B1 | Compare with correct tabular $t$ value |
| Claim is not correct | A1$\sqrt{}$ | Conclusion consistent with values (A.E.F.) |

**Total: [8]**

---
7 A greengrocer claims that his cabbages have a mean mass of more than 1.2 kg . In order to check his claim, he weighs 10 cabbages, chosen at random from his stock. The masses, in kg, are as follows.

$$\begin{array} { l l l l l l l l l l } 
1.26 & 1.24 & 1.17 & 1.23 & 1.18 & 1.25 & 1.19 & 1.20 & 1.21 & 1.18
\end{array}$$

Stating any assumption that you make, test at the $10 \%$ significance level whether the greengrocer's claim is supported by this evidence.

\hfill \mbox{\textit{CAIE FP2 2011 Q7 [8]}}