| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| 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 straightforward chi-squared goodness of fit test with given proportions. Students need to calculate expected frequencies (multiply proportions by total), compute chi-squared statistic, find degrees of freedom (4), and compare to critical value. It's slightly easier than average because the proportions are given explicitly, the calculation is routine, and it follows a standard template with no conceptual complications. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test5.06c Fit other distributions: discrete and continuous |
| Grade | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) |
| Proportion | \(4 \%\) | \(28 \%\) | \(52 \%\) | \(10 \%\) | \(6 \%\) |
| Grade | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) |
| Frequency | 9 | 71 | 136 | 21 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): Potassium has no effect on the quality of apple; \(H_1\): Potassium has an effect on the quality of apple | B1 | Both hypotheses in context. May use other wording e.g. The grading of apples remains the same |
| Expected values: \(A=9.6\), \(B=67.2\), \(C=124.8\), \(D=24.0\), \(E=14.4\) | M1A1 | A correct method to calculate expected values e.g. \(0.04\times240\); At least 3 expected values correct |
| \(\chi^2 = \sum\frac{(O-E)^2}{E} = \frac{(9-\text{"9.6"})^2}{\text{"9.6"}} + \ldots + \frac{(3-\text{"14.4"})^2}{\text{"14.4"}}\) or \(\chi^2 = \sum\frac{O^2}{E} - N = \frac{9^2}{\text{"9.6"}} + \ldots + \frac{3^2}{\text{"14.4"}} - 240\) | M1 | A correct method using their expected values to calculate \(\chi^2\). At least one correct, ft their expected values with an intention to add |
| \(= 10.657\ldots\) | A1 | awrt 10.7 |
| Degrees of freedom \(= 4\) | B1 | Degrees of freedom \(= 4\) (may be implied by 9.488) |
| \(\chi^2_{4,0.05} = 9.488\) | B1ft | 9.488 ft their DoF. If no DoF stated then this must be correct for their working |
| [Reject \(H_0\)] Data suggests that potassium may affect the distribution of the grades of apples or there is evidence that Andy's belief is incorrect | A1ft | ft their \(\chi^2\) value provided the 2nd M1 is awarded and CV. If no hypotheses or hypotheses wrong way round do not award. Must include the word 'apples' or 'belief' oe |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: Potassium has no effect on the quality of apple; $H_1$: Potassium has an effect on the quality of apple | B1 | Both hypotheses in context. May use other wording e.g. The grading of apples remains the same |
| Expected values: $A=9.6$, $B=67.2$, $C=124.8$, $D=24.0$, $E=14.4$ | M1A1 | A correct method to calculate expected values e.g. $0.04\times240$; At least 3 expected values correct |
| $\chi^2 = \sum\frac{(O-E)^2}{E} = \frac{(9-\text{"9.6"})^2}{\text{"9.6"}} + \ldots + \frac{(3-\text{"14.4"})^2}{\text{"14.4"}}$ or $\chi^2 = \sum\frac{O^2}{E} - N = \frac{9^2}{\text{"9.6"}} + \ldots + \frac{3^2}{\text{"14.4"}} - 240$ | M1 | A correct method using their expected values to calculate $\chi^2$. At least one correct, ft their expected values with an intention to add |
| $= 10.657\ldots$ | A1 | awrt 10.7 |
| Degrees of freedom $= 4$ | B1 | Degrees of freedom $= 4$ (may be implied by 9.488) |
| $\chi^2_{4,0.05} = 9.488$ | B1ft | 9.488 ft their DoF. If no DoF stated then this must be correct for their working |
| [Reject $H_0$] Data suggests that potassium may affect the distribution of the grades of apples **or** there is evidence that Andy's belief is incorrect | A1ft | ft their $\chi^2$ value provided the 2nd M1 is awarded and CV. If no hypotheses or hypotheses wrong way round do not award. Must include the word 'apples' or 'belief' oe |
---2. Andy has some apple trees. Over many years she has graded each apple from her trees as $A , B , C , D$ or $E$ according to the quality of the apple, with $A$ being the highest quality and $E$ being the lowest quality.
She knows that the proportion of apples in each grade produced by her trees is as follows.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Grade & $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
Proportion & $4 \%$ & $28 \%$ & $52 \%$ & $10 \%$ & $6 \%$ \\
\hline
\end{tabular}
\end{center}
Raj advises Andy to add potassium to the soil around her apple trees.
Andy believes that adding potassium will not affect the distribution of grades for the quality of the apples.
To test her belief Andy adds potassium to the soil around her apple trees. The following year she counts the number of apples in each grade. The number of apples in each grade is shown in the table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Grade & $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
Frequency & 9 & 71 & 136 & 21 & 3 \\
\hline
\end{tabular}
\end{center}
Test Andy's belief using a $5 \%$ level of significance. Show your working clearly, stating your hypotheses, expected frequencies and degrees of freedom.
2 continued\\
\hfill \mbox{\textit{Edexcel S3 2021 Q2 [8]}}