| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Expected frequencies partially provided |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with all calculations scaffolded: part (a) verifies given expected frequencies and contributions using routine formulas, while part (b) requires comparing a given test statistic to critical values. The question requires only procedural application of learned techniques with no problem-solving or conceptual insight, making it slightly easier than average. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Main use of their mobile phone | |||
| Internet | Texts | Phone calls | ||
| \multirow{3}{*}{Age} | Under 20 | 27 | 14 | 9 |
| From 20 to 40 | 32 | 34 | 29 | |
| Over 40 | 15 | 19 | 21 | |
| \multirow{2}{*}{} | Main use of their mobile phone | |||
| Internet | Texts | Phone calls | ||
| \multirow{3}{*}{Age} | Under 20 | 18.5 | 16.75 | 14.75 |
| From 20 to 40 | 35.15 | 31.825 | 28.025 | |
| Over 40 | 20.35 | 18.425 | 16.225 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Expected value \(= \frac{50 \times 74}{200} = 18.5\) | B1 cso | |
| \(\chi^2\) contribution \(= \frac{(27-18.5)^2}{18.5} = 3.905405405 = 3.91\) to 3sf | B1 cso | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): users age and main mobile phone use are independent/no association between users age and main mobile phone use; \(H_1\): users age and main mobile phone use are not independent/ some association between users age and main mobile phone use | B1 | |
| \(\nu = 4\) | B1 | |
| Critical value \(\chi^2 = 9.488\) | B1ft | |
| Test statistic is in critical region therefore significant evidence to reject \(H_0\) and accept \(H_1\). Evidence at 5% level that age and main phone use are not independent. | M1, A1ft | (5) (7 marks) |
**2(a)(i)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expected value $= \frac{50 \times 74}{200} = 18.5$ | B1 cso | |
| $\chi^2$ contribution $= \frac{(27-18.5)^2}{18.5} = 3.905405405 = 3.91$ to 3sf | B1 cso | (2) |
**2(a)(ii)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| | | |
**2(b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: users age and main mobile phone use are independent/no association between users age and main mobile phone use; $H_1$: users age and main mobile phone use are not independent/ some association between users age and main mobile phone use | B1 | |
| $\nu = 4$ | B1 | |
| Critical value $\chi^2 = 9.488$ | B1ft | |
| Test statistic is in critical region therefore significant evidence to reject $H_0$ and accept $H_1$. Evidence at 5% level that age and main phone use are not independent. | M1, A1ft | (5) (7 marks) |
**Guidance Notes:**
- 3rd B1 ft on their value of $\nu$
- M1 for attempt to compare test statistic and their critical value
- A1 ft on test statistic and critical value but must be comment in context. (A0 if hypotheses are the wrong way around)
---\begin{enumerate}
\item A survey asked a random sample of 200 people their age and the main use of their mobile phone.
\end{enumerate}
The results are shown in Table 1 below.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{3}{|c|}{Main use of their mobile phone} \\
\hline
& & Internet & Texts & Phone calls \\
\hline
\multirow{3}{*}{Age} & Under 20 & 27 & 14 & 9 \\
\hline
& From 20 to 40 & 32 & 34 & 29 \\
\hline
& Over 40 & 15 & 19 & 21 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
The data are to be used to test whether or not age and main use of their mobile phone are independent.
Table 2 shows the expected frequencies for each group, assuming people's age and main use of their mobile phone are independent.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{3}{|c|}{Main use of their mobile phone} \\
\hline
& & Internet & Texts & Phone calls \\
\hline
\multirow{3}{*}{Age} & Under 20 & 18.5 & 16.75 & 14.75 \\
\hline
& From 20 to 40 & 35.15 & 31.825 & 28.025 \\
\hline
& Over 40 & 20.35 & 18.425 & 16.225 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{center}
\end{table}
(a) For users under 20 choosing the Internet as the main use of their mobile phone,\\
(i) verify that the expected frequency is 18.5\\
(ii) show that the contribution to the $\chi ^ { 2 }$ test statistic is 3.91 to 3 significant figures.\\
(b) Given that the $\chi ^ { 2 }$ test statistic for the data is 9.893 to 3 decimal places, test at the $5 \%$ level of significance whether or not age and main use of their mobile phone are independent. State your hypotheses clearly.\\
\hfill \mbox{\textit{Edexcel S3 2014 Q2 [7]}}