| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×2 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a 2×2 contingency table. Students need to state hypotheses, calculate expected frequencies (straightforward with small numbers), compute the test statistic using the standard formula, compare to critical value, and conclude. It's slightly easier than average because the table is small (only 4 cells), the arithmetic is manageable, and it follows a well-rehearsed procedure with no conceptual surprises—typical S3 fare but requiring minimal problem-solving beyond applying the standard algorithm. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.06a Chi-squared: contingency tables |
| \cline { 2 - 3 } \multicolumn{1}{c|}{} | Good health | Poor health |
| Good diet | 86 | 8 |
| Poor diet | 91 | 15 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(H_0\): Diet and health are independent (or not associated). \(H_1\): Diet and health are not independent (or associated). | B1 | "diet" and "health" mentioned at least once. Use of correlation is B0. |
| Expected values: Good diet/Good health = 83.19, Good diet/Poor health = 10.81, Poor diet/Good health = 93.81, Poor diet/Poor health = 12.19 | M1; A1 | Attempt \(\frac{RT \times CT}{GT}\) with at least one correct to 1dp; all correct to 1dp. |
| \(\frac{(O-E)^2}{E}\) values: 0.095, 0.730, 0.084, 0.648. Total = 1.557 | M1 | Attempt at \(\frac{(O-E)^2}{E}\) or \(\frac{O^2}{E}\) with their values with 2 correct or 2 correct f.t. Allow 2sf. |
| \(\sum\frac{(O-E)^2}{E}\) or \(\sum\frac{O^2}{E} - 200 = 1.557\) | A1 | awrt 1.6 |
| \(\nu = (2-1)(2-1) = 1\) | B1 | 1 (may be implied) |
| \(\chi^2_1(5\%) = 3.841\) | B1f.t. | NB: may see \(\chi^2_3(5\%) = 7.815\) for f.t. NB: \(p\)-value 0.212 but scores B0B0 on its own |
| \(\chi^2 = 1.557\) does not lie in cr so insufficient evidence to reject \(H_0\) | M1 | For correct non-contextual statement linking their test statistic and their cv. |
| Diet and health are independent / no association between diet and health / doctor's belief is not supported | A1ft | Dependent on cv of 3.841 and 3rd M1. Must mention "diet" and "health" or "doctor". Condone "connection" or "relationship" but not "correlation". |
# Question 2:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $H_0$: Diet and health are independent (or not associated). $H_1$: Diet and health are not independent (or associated). | B1 | "diet" and "health" mentioned at least once. Use of correlation is B0. |
| Expected values: Good diet/Good health = 83.19, Good diet/Poor health = 10.81, Poor diet/Good health = 93.81, Poor diet/Poor health = 12.19 | M1; A1 | Attempt $\frac{RT \times CT}{GT}$ with at least one correct to 1dp; all correct to 1dp. |
| $\frac{(O-E)^2}{E}$ values: 0.095, 0.730, 0.084, 0.648. Total = 1.557 | M1 | Attempt at $\frac{(O-E)^2}{E}$ or $\frac{O^2}{E}$ with their values with 2 correct or 2 correct f.t. Allow 2sf. |
| $\sum\frac{(O-E)^2}{E}$ or $\sum\frac{O^2}{E} - 200 = 1.557$ | A1 | awrt 1.6 |
| $\nu = (2-1)(2-1) = 1$ | B1 | 1 (may be implied) |
| $\chi^2_1(5\%) = 3.841$ | B1f.t. | NB: may see $\chi^2_3(5\%) = 7.815$ for f.t. NB: $p$-value 0.212 but scores B0B0 on its own |
| $\chi^2 = 1.557$ does not lie in cr so insufficient evidence to reject $H_0$ | M1 | For correct non-contextual statement linking their test statistic and their cv. |
| Diet and health are independent / no association between diet and health / doctor's belief is not supported | A1ft | Dependent on cv of 3.841 and 3rd M1. Must mention "diet" and "health" or "doctor". Condone "connection" or "relationship" but not "correlation". |
---\begin{enumerate}
\item A doctor believes that the diet of her patients and their health are not independent.
\end{enumerate}
She takes a random sample of 200 patients and records whether they are in good health or poor health and whether they have a good diet or a poor diet. The results are summarised in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | }
\cline { 2 - 3 }
\multicolumn{1}{c|}{} & Good health & Poor health \\
\hline
Good diet & 86 & 8 \\
\hline
Poor diet & 91 & 15 \\
\hline
\end{tabular}
\end{center}
Stating your hypotheses clearly, test the doctor's belief using a $5 \%$ level of significance. Show your working for your test statistic and state your critical value clearly.
\hfill \mbox{\textit{Edexcel S3 2021 Q2 [9]}}