| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Paired sample t-test |
| Difficulty | Standard +0.3 This is a straightforward application of a paired t-test with clear data provided. Students need to calculate differences, find mean and standard deviation, compute the test statistic, and compare to critical value. While it requires multiple steps and knowledge of S4 content (Further Maths Statistics), the procedure is entirely standard with no novel problem-solving or conceptual challenges beyond routine application of the paired t-test formula. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Store | A | B | C | D | E | F | G | H |
| January ticket sales \(( \boldsymbol { \pounds } )\) | 1080 | 1639 | 710 | 1108 | 915 | 1066 | 1322 | 819 |
| July ticket sales \(( \boldsymbol { \pounds } )\) | 1113 | 1702 | 831 | 1048 | 861 | 1090 | 1303 | 852 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(d_i = \text{July} - \text{January}\): 33, 63, 121, −60, −54, 24, −19, 33 | B1 | All differences correct (allow January − July) |
| \(H_0: \mu_d = 0\), \(H_1: \mu_d > 0\) | B1 | Both hypotheses correct, where \(\mu_d\) is the population mean difference |
| \(\bar{d} = \frac{141}{8} = 17.625\) | M1 A1 | Method for mean of differences |
| \(S_{dd} = 33^2 + 63^2 + 121^2 + (-60)^2 + (-54)^2 + 24^2 + (-19)^2 + 33^2 - \frac{141^2}{8}\) | M1 | Correct method for \(S_{dd}\) |
| \(= 26557 - 2485.125 = 24071.875\) | A1 | Correct value |
| \(s_d^2 = \frac{24071.875}{7} = 3438.839...\) so \(s_d = 58.64...\) | ||
| \(t = \frac{17.625}{\frac{58.64...}{\sqrt{8}}} = \frac{17.625}{20.737...} = 0.8499...\) | M1 | Correct \(t\)-statistic method |
| Critical value \(t_7(5\%) = 1.895\) (one-tailed) | B1 | Correct critical value |
| Since \(0.850 < 1.895\), do not reject \(H_0\). No evidence at 5% significance level that mean sales are higher in July than January. | A1 | Correct conclusion in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| The differences in sales are normally distributed | B1 | Must reference the differences (not just sales) being normally distributed |
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $d_i = \text{July} - \text{January}$: 33, 63, 121, −60, −54, 24, −19, 33 | B1 | All differences correct (allow January − July) |
| $H_0: \mu_d = 0$, $H_1: \mu_d > 0$ | B1 | Both hypotheses correct, where $\mu_d$ is the population mean difference |
| $\bar{d} = \frac{141}{8} = 17.625$ | M1 A1 | Method for mean of differences |
| $S_{dd} = 33^2 + 63^2 + 121^2 + (-60)^2 + (-54)^2 + 24^2 + (-19)^2 + 33^2 - \frac{141^2}{8}$ | M1 | Correct method for $S_{dd}$ |
| $= 26557 - 2485.125 = 24071.875$ | A1 | Correct value |
| $s_d^2 = \frac{24071.875}{7} = 3438.839...$ so $s_d = 58.64...$ | | |
| $t = \frac{17.625}{\frac{58.64...}{\sqrt{8}}} = \frac{17.625}{20.737...} = 0.8499...$ | M1 | Correct $t$-statistic method |
| Critical value $t_7(5\%) = 1.895$ (one-tailed) | B1 | Correct critical value |
| Since $0.850 < 1.895$, do not reject $H_0$. No evidence at 5% significance level that mean sales are higher in July than January. | A1 | Correct conclusion in context |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| The differences in sales are normally distributed | B1 | Must reference the differences (not just sales) being normally distributed |
---\begin{enumerate}
\item The Sales Manager of a large chain of convenience stores is studying the sale of lottery tickets in her stores. She randomly selects 8 of her stores. From these stores she collects data for the total sales of lottery tickets in the previous January and July. The data are shown below
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Store & A & B & C & D & E & F & G & H \\
\hline
January ticket sales $( \boldsymbol { \pounds } )$ & 1080 & 1639 & 710 & 1108 & 915 & 1066 & 1322 & 819 \\
\hline
July ticket sales $( \boldsymbol { \pounds } )$ & 1113 & 1702 & 831 & 1048 & 861 & 1090 & 1303 & 852 \\
\hline
\end{tabular}
\end{center}
(a) Use a paired $t$-test to determine whether or not there is evidence, at the $5 \%$ level of significance, that the mean sales of lottery tickets in this chain's stores are higher in July than in January. You should state your hypotheses and show your working clearly.\\
(b) State what assumption the Sales Manager needs to make about the sales of lottery tickets in her stores for the test in part (a) to be valid.
\hfill \mbox{\textit{Edexcel S4 2015 Q1 [9]}}