| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard unbiased estimates calculation |
| Difficulty | Easy -1.2 This is a straightforward application of standard formulas for unbiased estimates (sample mean and variance with n-1 denominator) followed by a routine confidence interval calculation using the normal distribution. Both parts require only direct substitution into well-practiced formulas with no problem-solving or conceptual challenges. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{\Sigma x}{7} = \frac{34.7}{7} = 4.9571\) or \(4.96\) (3 sf); \((\Sigma x^2 = 175.15)\) | B1 | |
| \(\frac{7}{6}\left(\frac{\text{"175.15"}}{7} - \text{"4.9571"}^2\right)\) | M1 | |
| \(0.523\) (3 sf) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{'4.96'} \pm z \times \sqrt{\frac{0.523}{7}}\) | M1 | FT *their* mean and standard deviation |
| \(z = 1.96\) | B1 | |
| \(4.42\) to \(5.49\) (3 sf) | A1 | |
| 3 |
## Question 1:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{\Sigma x}{7} = \frac{34.7}{7} = 4.9571$ or $4.96$ (3 sf); $(\Sigma x^2 = 175.15)$ | B1 | |
| $\frac{7}{6}\left(\frac{\text{"175.15"}}{7} - \text{"4.9571"}^2\right)$ | M1 | |
| $0.523$ (3 sf) | A1 | |
| | **3** | |
**Part (b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{'4.96'} \pm z \times \sqrt{\frac{0.523}{7}}$ | M1 | **FT** *their* mean and standard deviation |
| $z = 1.96$ | B1 | |
| $4.42$ to $5.49$ (3 sf) | A1 | |
| | **3** | |1 The lengths, $X$ centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.\\
3.9\\
4.8\\
4.8\\
4.4\\
5.2\\
5.5\\
6.1
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the population mean and variance of $X$.\\
It is now given that the true value of the population variance of $X$ is 0.55 , and that $X$ has a normal distribution.
\item Find a 95\% confidence interval for the population mean of $X$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q1 [6]}}