| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | October |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Moderate -0.3 This is a standard chi-squared test of independence with straightforward calculations. Part (a) tests basic sampling knowledge, part (b) requires simple expected frequency calculations using row/column totals, and part (c) is routine hypothesis testing with most of the calculation already provided. The question requires no novel insight and follows a textbook template, making it slightly easier than average for A-level. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.01d Select/critique sampling: in context5.06a Chi-squared: contingency tables |
| Village | Number of households |
| A | 41 |
| B | 164 |
| C | 123 |
| D | 82 |
| \multirow{2}{*}{} | Age group | ||
| 18-49 | 50-69 | Older than 69 | |
| Listen to LSB | 130 | 162 | 65 |
| Do not listen to LSB | 78 | 98 | 62 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Label the houses in area A 1–41, area B 1–164, area C 1–123 and area D 1–82 | M1 | For suitable labelling of all four areas. E.g. for area A: \(1-41\) or \(0-40\) |
| Use random numbers to select a sample | M1 | For use of random numbers to select houses in each area |
| Simple random sample of 20 area A, 80 area B, 60 area C and 40 area D | A1 | For 20 A, 80B, 60C and 40D (dependent on 2nd M1 only) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{357\times260}{595}\) or \(\frac{238\times260}{595}\) | M1 | A correct method for finding one expected value |
| 156 and 104 | A1 | Correct answer for both values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{(162-\text{"156"})^2}{\text{"156"}} = \frac{3}{13} = 0.2307\ldots\) and \(\frac{(98-\text{"104"})^2}{\text{"104"}} = \frac{9}{26} = 0.3461\ldots\) | M1 | A correct method for finding both contributions to the \(\chi^2\) value |
| \(\chi^2 = 4.657 + \text{"0.2307"} + \text{"0.346"}\) | M1 | Adding the two values to 4.657 (may be implied by a full \(\chi^2\) calculation, do not ISW) |
| \(= 5.234\ldots\) | A1 | awrt 5.23 |
| \(\nu = (2-1)(3-1) = 2\) | B1 | 2 |
| \(\chi^2_2(0.05) = 5.991 \Rightarrow\) CR: \(\chi^2 > 5.991\) | B1ft | 5.991 or better ft their DoF |
| There is no evidence to suggest that there is an association between age and listening to LSB | dA1 | A correct contextual conclusion with words age and listening dependent on both M marks being awarded. NB if they give a \(p\) value of \(0.0730\ldots\) rather than the CV they can get M1M1B1B1B0A1 |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Label the houses in area A 1–41, area B 1–164, area C 1–123 and area D 1–82 | M1 | For suitable labelling of all four areas. E.g. for area A: $1-41$ or $0-40$ |
| Use random numbers to select a sample | M1 | For use of random numbers to select houses in each area |
| Simple random sample of 20 area A, 80 area B, 60 area C and 40 area D | A1 | For 20 A, 80B, 60C and 40D (dependent on 2nd M1 only) |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{357\times260}{595}$ or $\frac{238\times260}{595}$ | M1 | A correct method for finding one expected value |
| 156 and 104 | A1 | Correct answer for both values |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{(162-\text{"156"})^2}{\text{"156"}} = \frac{3}{13} = 0.2307\ldots$ and $\frac{(98-\text{"104"})^2}{\text{"104"}} = \frac{9}{26} = 0.3461\ldots$ | M1 | A correct method for finding both contributions to the $\chi^2$ value |
| $\chi^2 = 4.657 + \text{"0.2307"} + \text{"0.346"}$ | M1 | Adding the two values to 4.657 (may be implied by a full $\chi^2$ calculation, do not ISW) |
| $= 5.234\ldots$ | A1 | awrt 5.23 |
| $\nu = (2-1)(3-1) = 2$ | B1 | 2 |
| $\chi^2_2(0.05) = 5.991 \Rightarrow$ CR: $\chi^2 > 5.991$ | B1ft | 5.991 or better ft their DoF |
| There is no evidence to suggest that there is an association between age and listening to LSB | dA1 | A correct contextual conclusion with words age and listening dependent on both M marks being awarded. NB if they give a $p$ value of $0.0730\ldots$ rather than the CV they can get M1M1B1B1B0A1 |
---\begin{enumerate}
\item A local village radio station, LSB, decides to survey adults in its broadcasting area about the programmes it produces.\\
$L S B$ broadcasts to 4 villages $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D .\\
The number of households in each of the villages is given below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | }
\hline
Village & Number of households \\
\hline
A & 41 \\
\hline
B & 164 \\
\hline
C & 123 \\
\hline
D & 82 \\
\hline
\end{tabular}
\end{center}
LSB decides to take a stratified sample of 200 households.\\
(a) Explain how to select the households for this stratified sample.\\
(3)
One of the questions in the survey related to the age group of each member of the household and whether they listen to $L S B$. The data received are shown below.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\multirow{2}{*}{} & \multicolumn{3}{|c|}{Age group} \\
\hline
& 18-49 & 50-69 & Older than 69 \\
\hline
Listen to LSB & 130 & 162 & 65 \\
\hline
Do not listen to LSB & 78 & 98 & 62 \\
\hline
\end{tabular}
\end{center}
The data are to be used to determine whether or not there is an association between the age group and whether they listen to $L S B$.\\
(b) Calculate the expected frequencies for the age group 50-69 that\\
(i) listen to $L S B$\\
(ii) do not listen to $L S B$\\
(2)
Given that for the other 4 classes $\sum \frac { ( O - E ) ^ { 2 } } { E } = 4.657$ to 3 decimal places,\\
(c) test at the $5 \%$ level of significance, whether or not there is evidence of an association between age and listening to $L S B$. Show your working clearly, stating the degrees of freedom and the critical value.
\hfill \mbox{\textit{Edexcel S3 2021 Q4 [11]}}