| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a straightforward one-sample t-test with clear hypotheses (H₀: μ ≥ 2000 vs H₁: μ < 2000), small sample size requiring justification of the t-test, and standard calculations of sample mean, standard deviation, and test statistic. The 10% significance level and all necessary information are provided, making this a routine application of the t-test procedure with no conceptual challenges beyond standard S3 material. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\sigma^2\) unknown | B1 | 1 |
| (ii) \(H_0: \mu=2000\) (or \(\geq\)), \(H_1:\mu<2000\) | B1 | |
| \(\bar{x}=1958.2, s=115.57\) | B1B1 | or 1958,115.6 |
| EITHER: Test statistic \(=\frac{1958.2-2000}{115.57/2}\) | M1 | |
| \(=-0.7234\) | A1 | art \(-0.723\) |
| Critical value \(-1.638\) | B1 | |
| Test statistic not in CR, accept \(H_0\) | M1 | Or equivalent |
| Accept that specification is being met | A1 | Conclusion in context |
| OR: Critical region: \(\bar{x}-2000 < -t\frac{115.57/2}\) | M1 | |
| \(t=-1.638\) | B1 | |
| \(\bar{x} < 1905.2\) | A1 | art 1900 or 1910 |
| As above | M1A1 | 8 |
(i) $\sigma^2$ unknown | B1 | 1
(ii) $H_0: \mu=2000$ (or $\geq$), $H_1:\mu<2000$ | B1 |
$\bar{x}=1958.2, s=115.57$ | B1B1 | or 1958,115.6
EITHER: Test statistic $=\frac{1958.2-2000}{115.57/2}$ | M1 |
$=-0.7234$ | A1 | art $-0.723$
Critical value $-1.638$ | B1 |
Test statistic not in CR, accept $H_0$ | M1 | Or equivalent
Accept that specification is being met | A1 | Conclusion in context
OR: Critical region: $\bar{x}-2000 < -t\frac{115.57/2}$ | M1 |
$t=-1.638$ | B1 |
$\bar{x} < 1905.2$ | A1 | art 1900 or 1910
As above | M1A1 | 8 | Conclusion in context
---2 A particular type of engine used in rockets is designed to have a mean lifetime of at least 2000 hours. A check of four randomly chosen engines yielded the following lifetimes in hours.
$$\begin{array} { l l l l }
1896.4 & 2131.5 & 1903.3 & 1901.6
\end{array}$$
A significance test of whether engines meet the design is carried out. It may be assumed that lifetimes have a normal distribution.\\
(i) Give a reason why a $t$-test should be used.\\
(ii) Carry out the test at the $10 \%$ significance level.
\hfill \mbox{\textit{OCR S3 2006 Q2 [9]}}