| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared test of independence |
| Difficulty | Moderate -0.3 This is a standard chi-squared test for independence question requiring calculation of expected frequencies using the formula (row total × column total)/grand total, then completing a hypothesis test. The calculations are straightforward with clear row/column totals, and part (b) is scaffolded by providing most of the test statistic. This is slightly easier than average as it's a routine application of a well-practiced technique with no conceptual challenges. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Payment amount | |||
| Under £50 | £50 to £150 | Over £150 | ||
| \multirow{3}{*}{Payment method} | Cash | 23 | 19 | 18 |
| Bank card | 21 | 32 | 31 | |
| Mobile app | 16 | 39 | 41 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{60\times 60}{240}\) or \(\frac{60\times 84}{240}\) or \(\frac{60\times 96}{240}\) | M1 | For a correct method for finding one expected value |
| 15 and 21 and 24 | A2 | A1 for 2 correct answers or 1 correct and 3 values that sum to 60 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): There is no association between payment amount and payment method used; \(H_1\): There is an association between payment amount and payment method used | B1 | Both hypotheses correct. Must mention method and amount with payment at least once |
| \(\frac{(23-15)^2}{15}=4.2667\), \(\frac{(21-21)^2}{21}=0\), \(\frac{(16-24)^2}{24}=2.6667\) | M1 | For correct method finding all three contributions ft their part (a) |
| \(\chi^2 = 2.4048 + 4.2667 + 0 + 2.6667\) | M1 | For adding values to 2.4048 |
| \(= 9.3381\ldots\) awrt 9.34 | A1 | awrt 9.34 |
| \(\nu=(3-1)(3-1)=4\), \(\chi^2_4(0.05)=9.488 \Rightarrow\) CR: \(X^2 \geqslant 9.488\) | B1 B1ft | \(\nu=4\) implied by correct critical value of 9.488; 9.488 or better ft their DoF |
| Not in CR/Not significant. There is no evidence of an association between the payment amount and payment method used | dA1 | Dependent on both M marks. Must mention method and amount. Not "correlation" |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{60\times 60}{240}$ or $\frac{60\times 84}{240}$ or $\frac{60\times 96}{240}$ | M1 | For a correct method for finding one expected value |
| 15 and 21 and 24 | A2 | A1 for 2 correct answers or 1 correct and 3 values that sum to 60 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: There is no association between payment amount and payment method used; $H_1$: There is an association between payment amount and payment method used | B1 | Both hypotheses correct. Must mention method **and** amount with payment at least once |
| $\frac{(23-15)^2}{15}=4.2667$, $\frac{(21-21)^2}{21}=0$, $\frac{(16-24)^2}{24}=2.6667$ | M1 | For correct method finding all three contributions ft their part (a) |
| $\chi^2 = 2.4048 + 4.2667 + 0 + 2.6667$ | M1 | For adding values to 2.4048 |
| $= 9.3381\ldots$ awrt 9.34 | A1 | awrt 9.34 |
| $\nu=(3-1)(3-1)=4$, $\chi^2_4(0.05)=9.488 \Rightarrow$ CR: $X^2 \geqslant 9.488$ | B1 B1ft | $\nu=4$ implied by correct critical value of 9.488; 9.488 or better ft their DoF |
| Not in CR/Not significant. There is no evidence of an association between **the payment amount and payment method used** | dA1 | Dependent on both M marks. Must mention **method** and **amount**. Not "correlation" |
---\begin{enumerate}
\item A business accepts cash, bank cards or mobile apps as payment methods.
\end{enumerate}
The manager wishes to test whether or not there is an association between the payment amount and the payment method used.
The manager takes a random sample of 240 payments and records the payment amount and the payment method used.
The manager's results are shown in the table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{3}{|c|}{Payment amount} \\
\hline
& & Under £50 & £50 to £150 & Over £150 \\
\hline
\multirow{3}{*}{Payment method} & Cash & 23 & 19 & 18 \\
\hline
& Bank card & 21 & 32 & 31 \\
\hline
& Mobile app & 16 & 39 & 41 \\
\hline
\end{tabular}
\end{center}
Using these results,\\
(a) calculate the expected frequencies for the payment amount under $\pounds 50$ that\\
(i) use cash\\
(ii) use a bank card\\
(iii) use a mobile app
Given that for the other 6 classes $\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.4048$ to 4 decimal places,\\
(b) test, at the $5 \%$ level of significance, whether or not there is evidence for an association between the payment amount and the payment method used. You should state the hypotheses, the test statistic, the degrees of freedom and the critical value used for this test.
\hfill \mbox{\textit{Edexcel S3 2023 Q2 [10]}}