| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2022 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with clearly defined hypotheses, straightforward calculation of expected frequencies from a 2×3 contingency table, and routine comparison to critical values. While it requires multiple computational steps and proper statistical interpretation, it follows a well-practiced procedure with no conceptual surprises, making it slightly easier than the average A-level question. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Type of property | |||
| Bungalow | Flat | House | Total | |
| Sold within three months | 7 | 29 | 46 | 82 |
| Sold in more than three months | 9 | 19 | 15 | 43 |
| Total | 16 | 48 | 61 | 125 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): There is no association between type of property and the time taken to sell it; \(H_1\): There is an association between type of property and the time taken to sell it | B1 | Both hypotheses correct. Must mention "type of property" and "time taken" at least once (may be written in terms of independence) |
| Expected frequencies: Bungalow/Within 3 months = 10.496, Flat/Within 3 months = 31.488, House/Within 3 months = 40.016, Bungalow/More than 3 months = 5.504, Flat/More than 3 months = 16.512, House/More than 3 months = 20.984 | M1 A1 | Some attempt at \(\frac{\text{(Row Total)(Column Total)}}{\text{(Grand Total)}}\); All expected frequencies correct |
| \(\frac{(O-E)^2}{E}\) values: 1.1644, 0.1965, 0.8948, 2.2205, 0.3748, 1.7064 | dM1 A1 | Dependent on 1st M1 for at least 2 correct terms; at least 3 correct terms to 2dp or better |
| \(X^2 = \sum\frac{(O-E)^2}{E} = 6.557...\) awrt 6.56 | dM1 A1 | Dependent on 2nd M1; awrt 6.56 |
| \(\nu = (2-1)(3-1) = 2\) | B1 | \(\nu = 2\), can be implied by correct critical value of 5.991 |
| \(c^2_2(0.05) = 5.991 \Rightarrow\) CR: \(X^2 \geq 5.991\) | B1 | 5.991 |
| There is sufficient evidence to suggest there is an association between type of property and the time taken to sell it | A1 | Dependent on 3rd M1 and 3rd B1. Correct contextualised conclusion rejecting \(H_0\). Must mention type and time. Do not allow "correlation" |
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: There is no association between type of property and the time taken to sell it; $H_1$: There is an association between type of property and the time taken to sell it | B1 | Both hypotheses correct. Must mention "type of property" **and** "time taken" at least once (may be written in terms of independence) |
| Expected frequencies: Bungalow/Within 3 months = 10.496, Flat/Within 3 months = 31.488, House/Within 3 months = 40.016, Bungalow/More than 3 months = 5.504, Flat/More than 3 months = 16.512, House/More than 3 months = 20.984 | M1 A1 | Some attempt at $\frac{\text{(Row Total)(Column Total)}}{\text{(Grand Total)}}$; All expected frequencies correct |
| $\frac{(O-E)^2}{E}$ values: 1.1644, 0.1965, 0.8948, 2.2205, 0.3748, 1.7064 | dM1 A1 | Dependent on 1st M1 for at least 2 correct terms; at least 3 correct terms to 2dp or better |
| $X^2 = \sum\frac{(O-E)^2}{E} = 6.557...$ awrt 6.56 | dM1 A1 | Dependent on 2nd M1; awrt 6.56 |
| $\nu = (2-1)(3-1) = 2$ | B1 | $\nu = 2$, can be implied by correct critical value of 5.991 |
| $c^2_2(0.05) = 5.991 \Rightarrow$ CR: $X^2 \geq 5.991$ | B1 | 5.991 |
| There is sufficient evidence to suggest there is an **association** between **type** of property and the **time** taken to sell it | A1 | Dependent on 3rd M1 and 3rd B1. Correct contextualised conclusion rejecting $H_0$. Must mention **type** and **time**. Do not allow "correlation" |
---4. A manager at a large estate agency believes that the type of property affects the time taken to sell it.
A random sample of 125 properties sold is shown in the table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{2}{*}{} & \multicolumn{3}{|c|}{Type of property} & \\
\hline
& Bungalow & Flat & House & Total \\
\hline
Sold within three months & 7 & 29 & 46 & 82 \\
\hline
Sold in more than three months & 9 & 19 & 15 & 43 \\
\hline
Total & 16 & 48 & 61 & 125 \\
\hline
\end{tabular}
\end{center}
Test, at the $5 \%$ level of significance, whether there is evidence for an association between the type of property and the time taken to sell it. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test.\\
\hfill \mbox{\textit{Edexcel S3 2022 Q4 [10]}}