| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Unbiased estimates then CI |
| Difficulty | Standard +0.3 This is a straightforward confidence interval calculation using given summary statistics, followed by a simple comparison to the expected mean of 3. The mechanics are standard (calculate sample mean and variance, apply the formula), and the interpretation requires only checking if 3 lies within the interval. Slightly easier than average due to minimal conceptual demand and all necessary values being provided. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\text{est}(\mu) = 3.4\) | B1 | |
| \(\text{est}(\sigma^2) = \frac{100}{99}\left(\frac{1356}{100} - \text{'3.4'}^2\right) = 2.02(0202)\) | M1, A1 | \(\frac{1}{99}(1356 - 340^2/100)\) or \(200/99\) |
| \(z = 1.96\) | B1 | |
| \(3.4 \pm z \times \sqrt{\frac{2.020202'}{100}}\) | M1 | correct working only; allow from unbiased or biased variance |
| \(= 3.12\) to \(3.68\) (3 sf) | A1 | [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mean should be 3 | B1* | stated or implied |
| CI does not include 3; Machine probably not working properly | DB1\(\checkmark\) | [2] |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{est}(\mu) = 3.4$ | B1 | |
| $\text{est}(\sigma^2) = \frac{100}{99}\left(\frac{1356}{100} - \text{'3.4'}^2\right) = 2.02(0202)$ | M1, A1 | $\frac{1}{99}(1356 - 340^2/100)$ or $200/99$ |
| $z = 1.96$ | B1 | |
| $3.4 \pm z \times \sqrt{\frac{2.020202'}{100}}$ | M1 | correct working only; allow from unbiased or biased variance |
| $= 3.12$ to $3.68$ (3 sf) | A1 | [6] |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean should be 3 | B1* | stated or implied |
| CI does not include 3; Machine probably not working properly | DB1$\checkmark$ | [2] | $\checkmark$ their CI or evidence that… |
---6 A variable $X$ takes values $1,2,3,4,5$, and these values are generated at random by a machine. Each value is supposed to be equally likely, but it is suspected that the machine is not working properly. A random sample of 100 values of $X$, generated by the machine, gives the following results.
$$n = 100 \quad \Sigma x = 340 \quad \Sigma x ^ { 2 } = 1356$$
(i) Find a 95\% confidence interval for the population mean of the values generated by the machine.\\
(ii) Use your answer to part (i) to comment on whether the machine may be working properly.
\hfill \mbox{\textit{CAIE S2 2016 Q6 [8]}}