| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared with algebraic frequencies |
| Difficulty | Challenging +1.2 This is a chi-squared goodness of fit test with a twist - finding the range of y values that lead to rejection. It requires understanding of uniform distribution probabilities, calculating expected frequencies (8, 4, 8 from U(20)), setting up the chi-squared statistic as a function of y, and solving inequalities against the critical value. More challenging than routine chi-squared tests due to the algebraic manipulation required, but still a standard Further Maths technique. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Range | \(1 \leqslant x \leqslant 8\) | \(9 \leqslant x \leqslant 12\) | \(13 \leqslant x \leqslant 20\) |
| Observed frequency | 12 | \(y\) | \(28 - y\) |
| Answer | Marks | Guidance |
|---|---|---|
| Expected frequency | 16 | 8 |
| \(\displaystyle\sum\frac{(O-E)^2}{E}\): \(\dfrac{4^2}{16}\ \bigg | \ \dfrac{(y-8)^2}{8}\ \bigg | \ \dfrac{(12-y)^2}{16}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((0 \leq)\ y \leq 4\) or \(15 \leq y \leq 28\) | M1 A1 M1* depM1 B1 M1 M1 A2 [9] | M1: Method for expected frequencies. A1: Expected frequencies all correct. M1*: Use \(\displaystyle\sum\frac{(O-E)^2}{E}\). depM1: \(X^2 >\) CV. B1: Correct CV, ft on cells combined; 8+4+8: 3.841 (not 5.991). M1: Simplified 3-term quadratic. M1: Solve quadratic to obtain CVs (can be implied). A2: www. One number wrong or omitted: A1. Ignore \(0 \leq y\). Use of cc (Yates?) loses A marks only. SC: 8+4+8: M0A0 M1M1B1 M1M1A0A0 |
## Question 8:
| Expected frequency | 16 | 8 | 16 |
$\displaystyle\sum\frac{(O-E)^2}{E}$: $\dfrac{4^2}{16}\ \bigg|\ \dfrac{(y-8)^2}{8}\ \bigg|\ \dfrac{(12-y)^2}{16}$
$X^2 > 5.991 \Rightarrow$
$1 + \dfrac{(y-8)^2}{8} + \dfrac{(12-y)^2}{16} > 5.991$
$\Rightarrow 3y^2 - 56y + 192.144 > 0$
Critical values $4.53$ and $14.14$
$(0 \leq)\ y \leq 4$ or $15 \leq y \leq 28$ | **M1** A1 M1* depM1 B1 M1 M1 A2 [9] | M1: Method for expected frequencies. A1: Expected frequencies all correct. M1*: Use $\displaystyle\sum\frac{(O-E)^2}{E}$. depM1: $X^2 >$ CV. B1: Correct CV, ft on cells combined; 8+4+8: 3.841 (not 5.991). M1: Simplified 3-term quadratic. M1: Solve quadratic to obtain CVs (can be implied). A2: www. One number wrong or omitted: A1. Ignore $0 \leq y$. Use of cc (Yates?) loses A marks only. SC: 8+4+8: M0A0 M1M1B1 M1M1A0A0
The image you've shared appears to be only the **back cover/contact page** of an OCR mark scheme document — it contains only OCR's address, contact details, and legal information.
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Could you share the relevant mark scheme pages? I'd be happy to extract and format the content as requested once those pages are provided.8 The table shows the results of a random sample drawn from a population which is thought to have the distribution $\mathrm { U } ( 20 )$.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Range & $1 \leqslant x \leqslant 8$ & $9 \leqslant x \leqslant 12$ & $13 \leqslant x \leqslant 20$ \\
\hline
Observed frequency & 12 & $y$ & $28 - y$ \\
\hline
\end{tabular}
\end{center}
Find the range of values of $y$ for which the data are not consistent with the distribution at the $5 \%$ significance level.
\section*{END OF QUESTION PAPER}
\hfill \mbox{\textit{OCR Further Statistics AS 2018 Q8 [9]}}