| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Binomial |
| Difficulty | Standard +0.8 This is a standard chi-squared goodness of fit test with binomial distribution, requiring calculation of missing expected frequencies, appropriate grouping of cells, and hypothesis testing. While methodical, it involves multiple computational steps (finding binomial probabilities, calculating chi-squared statistic, determining degrees of freedom with grouping) and is typical of Further Maths statistics rather than standard A-level, placing it moderately above average difficulty. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Number of passes per group | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Observed frequency | 0 | 0 | 8 | 24 | 45 | 36 | 26 | 10 | 1 |
| Expected frequency | \(p\) | 1.180 | 6.193 | 18.579 | 34.836 | \(q\) | \(r\) | 13.437 | 2.519 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Using \(B(8, 0.6)\) | B1 | One correct |
| \(p = 0.098\), \(q = 41.804\), \(r = 31.353\) | B1 | Other two correct. If B0B0 scored, SCB1 if all 3 correct but not rounded to 3dp |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): the data fits the binomial distribution \(B(8, 0.6)\); \(H_1\): the data does not fit the binomial distribution \(B(8, 0.6)\) | B1 | |
| Add first 3 columns: O value 8, E value 7.471 AND last two columns: O value 11, E value 15.956 | M1 | Both |
| Chi-squared sum: \(0.0374 + 1.5817 + 2.9655 + 0.8058 + 0.9139 + 1.5394\) | M1 | Method must be seen |
| \(7.84\) | A1 | If M0 awarded SCB1 for 7.84 |
| Critical value, \(5df = 9.236\); \(7.84 < 9.236\); Accept \(H_0\) | M1 | Compare calculated value with 9.236 OR 13.36 if no columns added; 12.02 adding only last 2 columns; 10.64 adding only first 3 columns |
| There is insufficient evidence to reject the director's claim | A1 | Correct work only except possibly B1 for hypotheses, in context, level of uncertainty in language. 'prove' scores A0 |
| 6 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Using $B(8, 0.6)$ | B1 | One correct |
| $p = 0.098$, $q = 41.804$, $r = 31.353$ | B1 | Other two correct. If B0B0 scored, SCB1 if all 3 correct but not rounded to 3dp |
| | **2** | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: the data fits the binomial distribution $B(8, 0.6)$; $H_1$: the data does not fit the binomial distribution $B(8, 0.6)$ | B1 | |
| Add first 3 columns: O value 8, E value 7.471 AND last two columns: O value 11, E value 15.956 | M1 | Both |
| Chi-squared sum: $0.0374 + 1.5817 + 2.9655 + 0.8058 + 0.9139 + 1.5394$ | M1 | Method must be seen |
| $7.84$ | A1 | If M0 awarded SCB1 for 7.84 |
| Critical value, $5df = 9.236$; $7.84 < 9.236$; Accept $H_0$ | M1 | Compare calculated value with 9.236 OR 13.36 if no columns added; 12.02 adding only last 2 columns; 10.64 adding only first 3 columns |
| There is insufficient evidence to reject the director's claim | A1 | Correct work only except possibly B1 for hypotheses, in context, level of uncertainty in language. 'prove' scores A0 |
| | **6** | |2 An organisation runs courses to train students to become engineers. These students are taught in groups of 8 . The director of the organisation claims that on average $60 \%$ of the students in a group achieve a pass. A random sample of 150 groups of 8 students is chosen. The following table shows the observed frequencies together with some of the expected frequencies using the appropriate binomial distribution.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
Number of passes per group & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
Observed frequency & 0 & 0 & 8 & 24 & 45 & 36 & 26 & 10 & 1 \\
\hline
Expected frequency & $p$ & 1.180 & 6.193 & 18.579 & 34.836 & $q$ & $r$ & 13.437 & 2.519 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the values of $p , q$ and $r$ giving your answers correct to 3 decimal places.
\item Carry out a goodness of fit test, at the $10 \%$ significance level, to test whether there is evidence to reject the director's claim.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q2 [8]}}