| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Standard +0.3 This is a straightforward application of standard confidence interval formulas and normal distribution properties. Part (i) requires calculating a sample mean and applying the known formula for CI with known variance. Part (ii) involves finding probabilities for a sum of normals, which is routine given that students know the sum has variance 4σ². Both parts are textbook exercises with no problem-solving insight required, making this slightly easier than average. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(3 \pm (z \text{ or } t)\sqrt{(1/4)}\) | \(M1\) | Sample mean intended |
| With \(z = 1.645\) | \(B1\) | ART (2.18, 3.82) |
| \((2.1775, 3.8225)\) | \(A1\) | |
| \([3]\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(S \sim N(4\mu, 4)\) | \(M1\) | Or use distribution of sample mean with 2.75 |
| Use \(z=([11 - 4\mu]/^"2^"); 1.14/5 \& -2.14/5\) | \(M1 A1\) | |
| \(\phi(z)\) | \(M1\) | |
| \(= 0.016 \& 0.874\) | \(A1\) | |
| \([5]\) |
### (i)
Use $3 \pm (z \text{ or } t)\sqrt{(1/4)}$ | $M1$ | Sample mean intended
With $z = 1.645$ | $B1$ | ART (2.18, 3.82)
$(2.1775, 3.8225)$ | $A1$ |
| $[3]$ |
### (ii)
$S \sim N(4\mu, 4)$ | $M1$ | Or use distribution of sample mean with 2.75
Use $z=([11 - 4\mu]/^"2^"); 1.14/5 \& -2.14/5$ | $M1 A1$ |
$\phi(z)$ | $M1$ |
$= 0.016 \& 0.874$ | $A1$ |
| $[5]$ |The continuous random variable $U$ has a normal distribution with unknown mean $\mu$ and known variance 1. A random sample of four observations of $U$ gave the values
$3.9, 2.1, 4.6$ and $1.4$.
\begin{enumerate}[label=(\roman*)]
\item Calculate a $90\%$ confidence interval for $\mu$. [3]
\item The probability that the sum of four random observations of $U$ is less than 11 is denoted by $p$. For each of the end points of the confidence interval in part (i) calculate the corresponding value of $p$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR S3 2012 Q3 [8]}}