| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Standard +0.3 This is a straightforward t-distribution confidence interval question with small sample size. Part (i) requires standard application of the t-interval formula with calculation of sample mean and standard deviation. Part (ii) involves working backwards from interval width to find the confidence level, requiring understanding of the relationship between critical value and interval width. While it involves multiple steps and careful calculation, it's a routine application of standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\bar{x} = 62.5\) | B1 | |
| \(\hat{\sigma} = 0.294(392)\) | B1 | or var=\(13/150 = 0.0866\ldots\) Allow 0.087 for this mark. |
| \(t = 3.182\) | B1 | |
| \(\text{"62.5"} + \text{"3.182"} \times \text{"0.294"}/2\) | M1 | any t or z=1.96. \(\bar{x} \pm \frac{t\hat{\sigma}}{2}\) |
| \([62.032, 62.968]\) | A1 | Allow 4 or more dp. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| "0.294" \(t/2\) oe | B1ft | Must be t. z=B0. If t written, but z used B0 |
| \(= 0.241\) soi | B1 | |
| 1.637 | B1 | Allow 1.63, 1.638, 1.64 |
| 80% CAO | B1 |
## Question 4:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{x} = 62.5$ | B1 | |
| $\hat{\sigma} = 0.294(392)$ | B1 | or var=$13/150 = 0.0866\ldots$ Allow 0.087 for this mark. |
| $t = 3.182$ | B1 | |
| $\text{"62.5"} + \text{"3.182"} \times \text{"0.294"}/2$ | M1 | any t or z=1.96. $\bar{x} \pm \frac{t\hat{\sigma}}{2}$ |
| $[62.032, 62.968]$ | A1 | Allow 4 or more dp. |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| "0.294" $t/2$ oe | B1ft | Must be t. z=B0. If t written, but z used B0 |
| $= 0.241$ soi | B1 | |
| 1.637 | B1 | Allow 1.63, 1.638, 1.64 |
| 80% CAO | B1 | |
---4 A set of bathroom scales is known to operate with an error which is normally distributed. One morning a man weighs himself 4 times. The 4 values for his mass, in kg , which can be considered to be a random sample are as follows.
$$\begin{array} { l l l l }
62.6 & 62.8 & 62.1 & 62.5
\end{array}$$
(i) Find a $95 \%$ confidence interval for his mass. Give the end-points of the interval correct to 3 decimal places.\\
(ii) Based on these results, a $y \%$ confidence interval has width 0.482 . Find $y$.
\hfill \mbox{\textit{OCR S3 2015 Q4 [9]}}