| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2012 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a standard S3 hypothesis testing question covering one-sample t-test and two-sample t-test with variance equality check. All procedures are routine: calculate sample statistics, perform standard t-tests, and check variance assumption using F-test or variance ratio. While it requires multiple steps and careful execution, it involves no novel problem-solving—just systematic application of well-practiced techniques from the S3 syllabus. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu = 32, H_1: \mu < 32\) | \(B1\) | Or in words. Need population. [and 7(iii)(b)] |
| \(s_1^2 = (8586.19 - 289^2/10)/9 = (26.01)\) | \(B1\) | AEF eg s=5.1 |
| \(TS = (28.9 - 32)/(s/\sqrt{10}); s = \sqrt{26.01}\) | \(M1 A1\) | Allow M1A1 for (32-28.9)/(32-28.9) etc, but AEF eg s=5.1 Allow 1.922>1.833 (and A1 if earned.) |
| \(= -1.922\) | \(A1\) | A0 for TS=1.922 |
| Compare with \(-1.833\) and reject \(H_0\) | \(M1\) | Allow 1.922>1.833 (and A1 if earned.) |
| There is evidence at the 5% SL that spec not met | \(A1ft\) | Not too assertive. |
| \([7]\) |
| Answer | Marks | Guidance |
|---|---|---|
| No complaints if mean \(>\) spec | \(B1\) | Sensible reason |
| \([1]\) |
| Answer | Marks | Guidance |
|---|---|---|
| Equal population variances required for lives of batteries made in two factories | \(B1\) | Do not insist on 'population' |
| \(s_2^2 = (11290.95 - 363^2/12)/11\) | \(M1\) | Do not allow if 28.2 first seen in (b) |
| \(=28.2\),sample variances are close, so assumption valid | \(A1\) | |
| \([3]\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0:\mu_1 = \mu_2, H_1: \mu_1 \neq \mu_2\) | \(B1\) | Allow consistent use of '+' instead of '-' throughout |
| \(s^2 = (9 \times 26.01 + 11 \times 28.2)/20 = 27.2145\) | \(M1 A1\) | Or equivalent method. Allow from "26.01"/10+"28.2"/12. Allow M1 if 9 and/or 11 used instead of 10 and/or 12.Pooled or not. |
| \(TS = (28.9 - 363/12) /[s \sqrt{(10^{-1} + 12^{-1})^{1/2}}]\) | \(M1\) | Allow Alft for -0.607 from unpooled sample. |
| \(= -0.6044\) | \(A1\) | SC Allow for correct comparison with -1.325 for 1 tail test. |
| Compare correctly with \(-1.725\) and do not reject \(H_0\) | \(M1\) | Must have used +/- 1.725 |
| There is insufficient evidence (at the 10% sig level) that there is a difference in mean life of the batteries. | \(A1ft\) | |
| \([7]\) |
### (i)
$H_0: \mu = 32, H_1: \mu < 32$ | $B1$ | Or in words. Need population. [and 7(iii)(b)]
$s_1^2 = (8586.19 - 289^2/10)/9 = (26.01)$ | $B1$ | AEF eg s=5.1
$TS = (28.9 - 32)/(s/\sqrt{10}); s = \sqrt{26.01}$ | $M1 A1$ | Allow M1A1 for (32-28.9)/(32-28.9) etc, but AEF eg s=5.1 Allow 1.922>1.833 (and A1 if earned.)
$= -1.922$ | $A1$ | A0 for TS=1.922
Compare with $-1.833$ and reject $H_0$ | $M1$ | Allow 1.922>1.833 (and A1 if earned.)
There is evidence at the 5% SL that spec not met | $A1ft$ | Not too assertive.
| $[7]$ |
### (ii)
No complaints if mean $>$ spec | $B1$ | Sensible reason
| $[1]$ |
### (iii) (a)
Equal population variances required for lives of batteries made in two factories | $B1$ | Do not insist on 'population'
$s_2^2 = (11290.95 - 363^2/12)/11$ | $M1$ | Do not allow if 28.2 first seen in (b)
$=28.2$,sample variances are close, so assumption valid | $A1$ |
| $[3]$ |
### (iii) (b)
$H_0:\mu_1 = \mu_2, H_1: \mu_1 \neq \mu_2$ | $B1$ | Allow consistent use of '+' instead of '-' throughout
$s^2 = (9 \times 26.01 + 11 \times 28.2)/20 = 27.2145$ | $M1 A1$ | Or equivalent method. Allow from "26.01"/10+"28.2"/12. Allow M1 if 9 and/or 11 used instead of 10 and/or 12.Pooled or not.
$TS = (28.9 - 363/12) /[s \sqrt{(10^{-1} + 12^{-1})^{1/2}}]$ | $M1$ | Allow Alft for -0.607 from unpooled sample.
$= -0.6044$ | $A1$ | SC Allow for correct comparison with -1.325 for 1 tail test.
Compare correctly with $-1.725$ and do not reject $H_0$ | $M1$ | Must have used +/- 1.725
There is insufficient evidence (at the 10% sig level) that there is a difference in mean life of the batteries. | $A1ft$ |
| $[7]$ |The manufacturer's specification for batteries used in a certain electronic game is that the mean lifetime should be 32 hours. The manufacturer tests a random sample of 10 batteries made in Factory A, and the lifetimes ($x$ hours) are summarised by
$n = 10$, $\sum x = 289.0$ and $\sum x^2 = 8586.19$.
It may be assumed that the population of lifetimes has a normal distribution.
\begin{enumerate}[label=(\roman*)]
\item Carry out a one-tail test at the $5\%$ significance level of whether the specification is being met. [7]
\item Justify the use of a one-tail test in this context. [1]
\end{enumerate}
Batteries made with the same specification are also made in Factory B. The lifetimes of these batteries are also normally distributed. A random sample of 12 batteries from this factory was tested. The lifetimes are summarised by
$n = 12$, $\sum x = 363.0$ and $\sum x^2 = 11290.95$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item
\begin{enumerate}[label=(\alph*)]
\item State what further assumption must be made in order to test whether there is any difference in the mean lifetimes of batteries made at the two factories.
Use the data to comment on whether this assumption is reasonable. [3]
\item Carry out the test at the $10\%$ significance level. [7]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR S3 2012 Q7 [18]}}