| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2024 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Given ratios |
| Difficulty | Standard +0.3 This is a standard S3 chi-squared question with routine calculations. Part (a) tests goodness of fit with given ratios (straightforward expected frequencies from 10:5:2:3), part (b) requires basic expected frequency calculations for a contingency table, and part (c) asks for degrees of freedom using the standard formula. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.06a Chi-squared: contingency tables5.06b Fit prescribed distribution: chi-squared test5.06c Fit other distributions: discrete and continuous |
| \multirow{2}{*}{} | Ice cream flavour | |||||
| Vanilla | Chocolate | Strawberry | Other | Total | ||
| \multirow{3}{*}{Age} | Child | 95 | 25 | 13 | 25 | 158 |
| Teenager | 57 | 20 | 17 | 36 | 130 | |
| Adult | 36 | 50 | 10 | 16 | 112 | |
| Total | 188 | 95 | 40 | 77 | 400 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): favourite flavours occur in ratio \(10:5:2:3\); \(H_1\): do not occur in this ratio | B1 | Must state ratio or refer to "given ratio"; accept proportions statement |
| Expected values: Chocolate 200, Vanilla 100, Strawberry 40, Other 60 | M1 | At least 2 expected values correct; totals add to 200, 100, 40, 60 |
| \(\dfrac{(188-200)^2}{200} + \dfrac{(95-100)^2}{100} + \dfrac{(40-40)^2}{40} + \dfrac{(77-60)^2}{60}\) | M1 | Correct method for at least 2 flavours |
| \(\sum \dfrac{(O_i - E_i)^2}{E_i} = 5.786...\) awrt 5.79 | A1 | — |
| \(\nu = 3\) | B1 | Correct degrees of freedom |
| CV is \(7.815\) | B1ft | \(\nu=3\) gives 7.815; ft on 6 df gives 12.592 |
| \(5.79 < 7.815\), insufficient evidence to reject \(H_0\) | M1 | Independent of hypotheses; ft their \(\chi^2\) and CV |
| No evidence that flavours do not occur in given ratio | A1ft | Dependent on all previous method marks; needs flavour and ratio |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{188 \times 130}{400}\) or \(\dfrac{112 \times 95}{400}\) | M1 | Correct method for one value |
| \(61.1\) and \(26.6\) | A1 | Both correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6\) | B1 | cao |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: favourite flavours occur in ratio $10:5:2:3$; $H_1$: do not occur in this ratio | B1 | Must state ratio or refer to "given ratio"; accept proportions statement |
| Expected values: Chocolate 200, Vanilla 100, Strawberry 40, Other 60 | M1 | At least 2 expected values correct; totals add to 200, 100, 40, 60 |
| $\dfrac{(188-200)^2}{200} + \dfrac{(95-100)^2}{100} + \dfrac{(40-40)^2}{40} + \dfrac{(77-60)^2}{60}$ | M1 | Correct method for at least 2 flavours |
| $\sum \dfrac{(O_i - E_i)^2}{E_i} = 5.786...$ awrt **5.79** | A1 | — |
| $\nu = 3$ | B1 | Correct degrees of freedom |
| CV is $7.815$ | B1ft | $\nu=3$ gives 7.815; ft on 6 df gives 12.592 |
| $5.79 < 7.815$, insufficient evidence to reject $H_0$ | M1 | Independent of hypotheses; ft their $\chi^2$ and CV |
| No evidence that **flavours** do not occur in **given ratio** | A1ft | Dependent on all previous method marks; needs **flavour** and **ratio** |
## Part (b)(i)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{188 \times 130}{400}$ or $\dfrac{112 \times 95}{400}$ | M1 | Correct method for one value |
| $61.1$ and $26.6$ | A1 | Both correct |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6$ | B1 | cao |
---\begin{enumerate}
\item The manager of a company making ice cream believes that the proportions of people in the population who prefer vanilla, chocolate, strawberry and other are in the ratio $10 : 5 : 2 : 3$
\end{enumerate}
The manager takes a random sample of 400 customers and records their age and favourite ice cream flavour. The results are shown in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{4}{|c|}{Ice cream flavour} & \\
\hline
& & Vanilla & Chocolate & Strawberry & Other & Total \\
\hline
\multirow{3}{*}{Age} & Child & 95 & 25 & 13 & 25 & 158 \\
\hline
& Teenager & 57 & 20 & 17 & 36 & 130 \\
\hline
& Adult & 36 & 50 & 10 & 16 & 112 \\
\hline
& Total & 188 & 95 & 40 & 77 & 400 \\
\hline
\end{tabular}
\end{center}
(a) Use the data in the table to test, at the $5 \%$ level of significance, the manager's belief. You should state your hypotheses, test statistic, critical value and conclusion clearly.
A researcher wants to investigate whether or not there is a relationship between the age of a customer and their favourite ice cream flavour. In order to test whether favourite ice cream flavour and age are related, the researcher plans to carry out a $\chi ^ { 2 }$ test.\\
(b) Use the table to calculate expected frequencies for the group\\
(i) teenagers whose favourite ice cream flavour is vanilla,\\
(ii) adults whose favourite ice cream flavour is chocolate.\\
(c) Write down the number of degrees of freedom for this $\chi ^ { 2 }$ test.
\hfill \mbox{\textit{Edexcel S3 2024 Q4 [11]}}