| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Recover sample stats from CI |
| Difficulty | Standard +0.3 This question tests understanding of confidence intervals using the t-distribution with straightforward algebraic manipulation to find sample statistics from interval endpoints, plus interpretation. Part (i) requires simple averaging and rearranging the CI formula (routine A-level technique), part (ii) is standard recall (normality assumption), and part (iii) requires basic interpretation that the population mean lies outside the CI. While it involves Further Maths content (t-distribution), the actual mathematical demands are modest—no complex calculations or novel insights required, making it slightly easier than average. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(\bar{x} = \frac{1}{2}(61.21 + 64.39) = 62.8\) | M1 A1 | Find sample mean |
| \(\bar{x} \pm ts/\sqrt{n}\) for any \(t\) | M1 | Use confidence interval formula |
| \(t_{24,099} = 2.492\) | A1 | Use correct tabular \(t\) |
| \(s = 1.59 \times 5 / 2.492 = 3.19\) | A1 | Calculate standard deviation; Part total: 5 |
| (ii) Population has normal distribution | B1 | State assumption (A.E.F.); Part total: 1 |
| (iii) 72 exceeds upper limit of interval | *B1 | State valid reason (A.E.F.) |
| Yes, it does reduce pulse rate | B1 | State conclusion (A.E.F., dep *B1); Part total: 2 |
| Total: 8 |
## Question 7:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** $\bar{x} = \frac{1}{2}(61.21 + 64.39) = 62.8$ | M1 A1 | Find sample mean |
| $\bar{x} \pm ts/\sqrt{n}$ for any $t$ | M1 | Use confidence interval formula |
| $t_{24,099} = 2.492$ | A1 | Use correct tabular $t$ |
| $s = 1.59 \times 5 / 2.492 = 3.19$ | A1 | Calculate standard deviation; **Part total: 5** |
| **(ii)** Population has normal distribution | B1 | State assumption (A.E.F.); **Part total: 1** |
| **(iii)** 72 exceeds upper limit of interval | *B1 | State valid reason (A.E.F.) |
| Yes, it does reduce pulse rate | B1 | State conclusion (A.E.F., dep *B1); **Part total: 2** |
| | | **Total: 8** |
---7 The pulse rate of each member of a random sample of 25 adult UK males who exercise for a given period each week is measured in beats per minute. A $98 \%$ confidence interval for the mean pulse rate, $\mu$ beats per minute, for all such UK males was calculated as $61.21 < \mu < 64.39$, based on a $t$-distribution.\\
(i) Calculate the sample mean pulse rate and the standard deviation used in the calculation.\\
(ii) State an assumption necessary for the validity of the confidence interval.\\
(iii) The mean pulse rate for all UK males is 72 beats per minute. State, giving a reason, if it can be concluded that, on average, UK males who exercise have a reduced pulse rate.
\hfill \mbox{\textit{CAIE FP2 2008 Q7 [8]}}