CAIE FP2 2018 June — Question 7 7 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2018
SessionJune
Marks7
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 all steps clearly signaled: calculate sample mean and standard deviation from given data, set up hypotheses for μ=94, compute t-statistic, compare to critical value at 10% significance. While it requires careful arithmetic and knowledge of the t-test procedure, it's a standard textbook application with no conceptual subtleties or novel problem-solving required.
Spec5.05c Hypothesis test: normal distribution for population mean

7 A large number of athletes are taking part in a competition. The masses, in kg , of a random sample of 7 athletes are as follows. $$\begin{array} { l l l l l l l } 98.1 & 105.0 & 92.2 & 89.8 & 99.9 & 95.4 & 101.2 \end{array}$$ Assuming that masses are normally distributed, test, at the \(10 \%\) significance level, whether the mean mass of athletes in this competition is equal to 94 kg .

Question 7:
AnswerMarks Guidance
\(\bar{x} = 681.6/7 = 97.37\) (allow 97.4 for this B1)B1 Find sample mean
\(s^2 = (66\,536.1 - 681.6^2/7)/6 \quad [= 27.96 \text{ or } 5.287^2]\)M1 Estimate population variance (allow biased: \(23.96\) or \(4.895^2\))
\(H_0: \mu = 94,\ H_1: \mu \neq 94\)B1 (AEF) State hypotheses (B0 for \(\bar{x}\ ...\))
\(t = (\bar{x} - 94)/(s/\sqrt{7}) = 1.69\)M1 A1 Find value of \(t\)
\(t_{7,\,0.95} = 1.94[3]\)B1 State or use correct tabular \(t\)-value (or can compare \(\bar{x}\) with \(94 + 3.88 = 97.88\))
[Accept \(H_0\):] Mean mass is equal to 94 kg\(\text{B1}\sqrt{}\) (AEF) Consistent conclusion (FT on both \(t\)-values)
## Question 7:

| $\bar{x} = 681.6/7 = 97.37$ (allow 97.4 for this B1) | B1 | Find sample mean |
|---|---|---|
| $s^2 = (66\,536.1 - 681.6^2/7)/6 \quad [= 27.96 \text{ or } 5.287^2]$ | M1 | Estimate population variance (allow biased: $23.96$ or $4.895^2$) |
| $H_0: \mu = 94,\ H_1: \mu \neq 94$ | B1 (AEF) | State hypotheses (B0 for $\bar{x}\ ...$) |
| $t = (\bar{x} - 94)/(s/\sqrt{7}) = 1.69$ | M1 A1 | Find value of $t$ |
| $t_{7,\,0.95} = 1.94[3]$ | B1 | State or use correct tabular $t$-value (or can compare $\bar{x}$ with $94 + 3.88 = 97.88$) |
| [Accept $H_0$:] Mean mass is equal to 94 kg | $\text{B1}\sqrt{}$ (AEF) | Consistent conclusion (FT on both $t$-values) |

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7 A large number of athletes are taking part in a competition. The masses, in kg , of a random sample of 7 athletes are as follows.

$$\begin{array} { l l l l l l l } 
98.1 & 105.0 & 92.2 & 89.8 & 99.9 & 95.4 & 101.2
\end{array}$$

Assuming that masses are normally distributed, test, at the $10 \%$ significance level, whether the mean mass of athletes in this competition is equal to 94 kg .\\

\hfill \mbox{\textit{CAIE FP2 2018 Q7 [7]}}