| 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}
\begin{enumerate}[label=(\alph*)]
\item 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.
\item Use the table to calculate expected frequencies for the group\\
\begin{enumerate}[label=(\roman*)]
\item teenagers whose favourite ice cream flavour is vanilla,
\item adults whose favourite ice cream flavour is chocolate.
\end{enumerate}
\item Write down the number of degrees of freedom for this $\chi ^ { 2 }$ test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2024 Q4 [11]}}