| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| 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 +0.3 This is a straightforward chi-squared goodness of fit test with a binomial distribution. Part (a) requires simple calculation of missing expected frequencies using B(8,0.15) and ensuring totals sum to 150. Part (b) is a standard hypothesis test procedure: combine cells to meet the rule of 5, calculate test statistic, compare to critical value. All steps are routine applications of learned procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Number of unripe pears per pack | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
| Observed frequency | 35 | 48 | 43 | 15 | 6 | 3 | 0 |
| Expected frequency | 40.874 | \(p\) | 35.641 | 12.579 | 2.775 | 0.392 | \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 0 | 1 |
| O | 35 | 48 |
| E | 40.874 | 57.704 |
| \(p = 57.704\) | B1 | 3 dp or better |
| \(q = 0.035\) or \(0.036\) | B1 | Accept numbers which round to 0.035 or 0.036 |
| Answer | Marks | Guidance |
|---|---|---|
| Combine frequencies less than 5: last 4 columns give \(24/15.7812\) | M1 | Allow last 2 or 3 columns combined for this M1 |
| \(\sum\frac{(O-E)^2}{E} = 0.8444 + 1.6319 + 1.5199 + 4.2803\) | M1 | |
| \(8.28\) | A1 | Accept \(8.27 - 8.28\). SC: 25.60 if values not combined, scores M1A0. SC: 14.96 if last 3 combined, scores M1A0. SC: if last 2 combined, scores M1A0 |
| \(H_0\): B(8, 0.15) fits the data | B1 | Must mention distribution and data |
| 3 degrees of freedom, tabular value \(= 7.815\). \(8.28 > 7.815\). Reject \(H_0\). | M1 | Compare their value with 7.815 and correct FT conclusion. 3 columns combined compared with 9.488 and correct conclusion |
| Insufficient evidence to support manager's claim. | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used |
## Question 3(a):
| $x$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 or more |
|---|---|---|---|---|---|---|---|
| O | 35 | 48 | 43 | 15 | 6 | 3 | 0 |
| E | 40.874 | 57.704 | 35.641 | 12.579 | 2.775 | 0.392 | 0.0352 |
$p = 57.704$ | B1 | 3 dp or better |
$q = 0.035$ or $0.036$ | B1 | Accept numbers which round to 0.035 or 0.036 |
## Question 3(b):
Combine frequencies less than 5: last 4 columns give $24/15.7812$ | M1 | Allow last 2 or 3 columns combined for this M1 |
$\sum\frac{(O-E)^2}{E} = 0.8444 + 1.6319 + 1.5199 + 4.2803$ | M1 | |
$8.28$ | A1 | Accept $8.27 - 8.28$. SC: 25.60 if values not combined, scores M1A0. SC: 14.96 if last 3 combined, scores M1A0. SC: if last 2 combined, scores M1A0 |
$H_0$: B(8, 0.15) fits the data | B1 | Must mention distribution and data |
3 degrees of freedom, tabular value $= 7.815$. $8.28 > 7.815$. Reject $H_0$. | M1 | Compare their value with 7.815 and correct FT conclusion. 3 columns combined compared with 9.488 and correct conclusion |
Insufficient evidence to support manager's claim. | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used |3 A supermarket sells pears in packs of 8 . Some of the pears in a pack may not be ripe, and the supermarket manager claims that the number of unripe pears in a pack can be modelled by the distribution $\mathrm { B } ( 8,0.15 )$.
A random sample of 150 packs was selected and the number of unripe pears in each pack was recorded. The following table shows the observed frequencies together with some of the expected frequencies using the manager's binomial distribution.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Number of unripe pears per pack & 0 & 1 & 2 & 3 & 4 & 5 & $\geqslant 6$ \\
\hline
Observed frequency & 35 & 48 & 43 & 15 & 6 & 3 & 0 \\
\hline
Expected frequency & 40.874 & $p$ & 35.641 & 12.579 & 2.775 & 0.392 & $q$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the values of $p$ and $q$.
\item Carry out a goodness of fit test, at the $5 \%$ significance level, to test whether the manager's claim is justified.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q3 [8]}}