| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2013 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Find k or boundary value from test |
| Difficulty | Standard +0.3 This is a standard two-sample t-test with summary statistics requiring calculation of means, variances, pooled variance, and test statistic. Part (ii) adds a slight twist by asking for a critical difference value, but this is a routine manipulation of the confidence interval formula. The question is slightly above average difficulty due to the two parts and computational demands, but follows a well-practiced procedure with no conceptual surprises for S3 students. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Sample size | \(\sum t\) | \(\sum t ^ { 2 }\) | |
| Machine \(A\) | 10 | 221.4 | 4920.9 |
| Machine \(B\) | 8 | 199.2 | 4980.3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (Assumes the populations of machine) times have equal variances | B1 | unless explicitly samples |
| \(H_0: \mu_A = \mu_B,\ H_1: \mu_A \neq \mu_B\) | B1 | For both. Accept in words, provided 'population' used |
| Pooled estimate \(= s^2\) | ||
| \(= [4920.9 - 221.4^2/10 + 4980.3 - 199.2^2/8]/16 = 39.324/16\) | M1A1 | A1 for 39.324 or 16. Not pooled M0A0A0 here |
| \(= 2.45775\) | A1 | |
| SD of \(\bar{t}_B - \bar{t}_A = s\sqrt{10^{-1} + 8^{-1}}\) | ||
| \(= 0.74364\) | A1 | A0 for 0.757 |
| \(1\%\ \text{CV} = 2.921\) | B1 | allow B1 for 2.921 |
| \(\bar{t}_B - \bar{t}_A = 2.76\) or \(24.9 - 22.14\) | B1 | allow B1 for 2.76 etc |
| Test statistic \(T = 2.76/\text{SD}\) | M1 | Needs denominator \((s)\ \left(\frac{s}{10}, \frac{s}{8}\right), \left(\frac{s}{9}, \frac{s}{7}\right), \frac{s}{18}, \frac{s}{16}\) for M1 |
| \(= 3.711\) | A1 | allow A1ft for 3.645 |
| \(T > 2.921\) so reject \(H_0\) | M1 | ft TS, CV even if 1-tail and/or \(z\). Consistent signs |
| There is sufficient evidence that the population mean packing times differ | A1 | ft TS only. 'Not pooled' can score B1B1M0A0A0A0B1M1A1M1A1 max 8/12 |
| [12] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| CV \(2.583\) | B1 | |
| \((2.76 - c)/\text{SD} \geq \text{CV}\) | M1 | Allow \(=,\ \leq\); must be sd from (i) |
| \(c \leq 0.8392\) or largest \(\approx 0.839\) | A1 | NOT isw |
| [3] |
# Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| (Assumes the populations of machine) times have equal variances | B1 | unless explicitly samples |
| $H_0: \mu_A = \mu_B,\ H_1: \mu_A \neq \mu_B$ | B1 | For both. Accept in words, provided 'population' used |
| Pooled estimate $= s^2$ | | |
| $= [4920.9 - 221.4^2/10 + 4980.3 - 199.2^2/8]/16 = 39.324/16$ | M1A1 | A1 for 39.324 or 16. Not pooled M0A0A0 here |
| $= 2.45775$ | A1 | |
| SD of $\bar{t}_B - \bar{t}_A = s\sqrt{10^{-1} + 8^{-1}}$ | | |
| $= 0.74364$ | A1 | A0 for 0.757 |
| $1\%\ \text{CV} = 2.921$ | B1 | allow B1 for 2.921 |
| $\bar{t}_B - \bar{t}_A = 2.76$ or $24.9 - 22.14$ | B1 | allow B1 for 2.76 etc |
| Test statistic $T = 2.76/\text{SD}$ | M1 | Needs denominator $(s)\ \left(\frac{s}{10}, \frac{s}{8}\right), \left(\frac{s}{9}, \frac{s}{7}\right), \frac{s}{18}, \frac{s}{16}$ for M1 |
| $= 3.711$ | A1 | allow A1ft for 3.645 |
| $T > 2.921$ so reject $H_0$ | M1 | ft TS, CV even if 1-tail and/or $z$. Consistent signs |
| There is sufficient evidence that the population mean packing times differ | A1 | ft TS only. 'Not pooled' can score B1B1M0A0A0A0B1M1A1M1A1 max 8/12 |
| | **[12]** | |
---
# Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| CV $2.583$ | B1 | |
| $(2.76 - c)/\text{SD} \geq \text{CV}$ | M1 | Allow $=,\ \leq$; must be sd from (i) |
| $c \leq 0.8392$ or largest $\approx 0.839$ | A1 | NOT isw |
| | **[3]** | |
The image you've shared appears to be only the **back cover/contact page** of an OCR mark scheme document — it contains only OCR's contact details, address, and administrative information.
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To extract mark scheme content, please share the **interior pages** of the document that contain the actual marking grid/table with questions and answers.7 Two machines $A$ and $B$ both pack cartons in a factory. The mean packing times are compared by timing the packing of 10 randomly chosen cartons from machine $A$ and 8 randomly chosen cartons from machine $B$. The times, $t$ seconds, taken to pack these cartons are summarised below.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& Sample size & $\sum t$ & $\sum t ^ { 2 }$ \\
\hline
Machine $A$ & 10 & 221.4 & 4920.9 \\
\hline
Machine $B$ & 8 & 199.2 & 4980.3 \\
\hline
\end{tabular}
\end{center}
The packing times have independent normal distributions.\\
(i) Stating a necessary assumption, carry out a test, at the $1 \%$ significance level, of whether the population mean packing times differ for the two machines.\\
(ii) Find the largest possible value of the constant $c$ for which there is evidence at the $1 \%$ significance level that $\mu _ { B } - \mu _ { A } > c$, where $\mu _ { B }$ and $\mu _ { A }$ denote the respective population mean packing times in seconds.
\hfill \mbox{\textit{OCR S3 2013 Q7 [15]}}