| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question requiring standard formula application (part i), basic interpretation of whether a claimed value lies within the interval (part ii), and understanding sampling variability (part iii). All parts are routine for S2 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(29.6 \pm z \times \frac{1.0}{\sqrt{65}}\) | M1 | Allow any value of \(z\) |
| \(29.6 \pm 2.576 \times \frac{1.0}{\sqrt{65}}\) | B1 | For 2.576 seen |
| \((29.6 \pm 0.3195)\) | ||
| \((29.3, 29.9)\) (3 sfs) | A1 [3] | Allow any brackets or none, but cwo |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| CI does not include 30 | B1ft | 30 seen or implied |
| Claim not supported or not justified or probably not true | B1ft [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| CI is a variable | B1 [1] | Allow "Sample mean diff" (not population mean) |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $29.6 \pm z \times \frac{1.0}{\sqrt{65}}$ | M1 | Allow any value of $z$ |
| $29.6 \pm 2.576 \times \frac{1.0}{\sqrt{65}}$ | B1 | For 2.576 seen |
| $(29.6 \pm 0.3195)$ | | |
| $(29.3, 29.9)$ (3 sfs) | A1 [3] | Allow any brackets or none, but cwo |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| CI does not include 30 | B1ft | 30 seen or implied |
| Claim not supported or not justified or probably not true | B1ft [2] | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| CI is a variable | B1 [1] | Allow "Sample mean diff" (not population mean) |
---3 The masses of sweets produced by a machine are normally distributed with mean $\mu$ grams and standard deviation 1.0 grams. A random sample of 65 sweets produced by the machine has a mean mass of 29.6 grams.\\
(i) Find a $99 \%$ confidence interval for $\mu$.
The manufacturer claims that the machine produces sweets with a mean mass of 30 grams.\\
(ii) Use the confidence interval found in part (i) to draw a conclusion about this claim.\\
(iii) Another random sample of 65 sweets produced by the machine is taken. This sample gives a $99 \%$ confidence interval that leads to a different conclusion from that found in part (ii). Assuming that the value of $\mu$ has not changed, explain how this can be possible.
\hfill \mbox{\textit{CAIE S2 2010 Q3 [6]}}