| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Confidence intervals for population mean |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question requiring standard formulas for unbiased estimates, constructing a confidence interval using t-distribution, and solving for sample size. All steps are routine applications of S2 techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02g Calculate mean and standard deviation5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(s^2 = \frac{1}{29}\left(33141816 - \frac{31500^2}{30}\right) = 2304\) | B1, M1, A1 | Correct mean. Correct formula with 29 in denom. Correct answer. |
| (ii) \(1050 \pm 2.326 \times \frac{48}{\sqrt{30}} = (1030, 1070)\) | M1, B1, A1ℜ | Correct shape with \(\sqrt{30}\) in denom. 2.326 seen. or equivalent, ft on their mean and variance. |
| Answer | Marks | Guidance |
|---|---|---|
| \(n = 246\) | M1*, M1dep, A1 | Correct form of LHS of equation/inequality involving 1.96, 48, \(\sqrt{n}\). Equated to 6 and attempt to solve (accept factor of 2 errors). Correct answer. SR If M0 M0 scored but only error is in z value score M1. |
**(i)** $\bar{x} = 1050$
$s^2 = \frac{1}{29}\left(33141816 - \frac{31500^2}{30}\right) = 2304$ | B1, M1, A1 | Correct mean. Correct formula with 29 in denom. Correct answer.
**(ii)** $1050 \pm 2.326 \times \frac{48}{\sqrt{30}} = (1030, 1070)$ | M1, B1, A1ℜ | Correct shape with $\sqrt{30}$ in denom. 2.326 seen. or equivalent, ft on their mean and variance.
**(iii)** $1.96 \times \frac{48}{\sqrt{n}} = 6$
$n = 246$ | M1*, M1dep, A1 | Correct form of LHS of equation/inequality involving 1.96, 48, $\sqrt{n}$. Equated to 6 and attempt to solve (accept factor of 2 errors). Correct answer. SR If M0 M0 scored but only error is in z value score M1.
---6 The daily takings, $\$ x$, for a shop were noted on 30 randomly chosen days. The takings are summarised by $\Sigma x = 31500 , \Sigma x ^ { 2 } = 33141816$.\\
(i) Calculate unbiased estimates of the population mean and variance of the shop's daily takings.\\
(ii) Calculate a $98 \%$ confidence interval for the mean daily takings.
The mean daily takings for a random sample of $n$ days is found.\\
(iii) Estimate the value of $n$ for which it is approximately $95 \%$ certain that the sample mean does not differ from the population mean by more than $\$ 6$.
\hfill \mbox{\textit{CAIE S2 2007 Q6 [9]}}