| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Uniform |
| Difficulty | Moderate -0.3 This is a straightforward chi-squared goodness of fit test with standard bookwork. Part (a) is simple arithmetic verification, part (b) requires comparing a given test statistic to critical values (calculation already done), and part (c) asks for interpretation of the data. All steps are routine applications of the chi-squared test procedure with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Integer | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| O | 7 | 8 | 20 | 8 | 7 | 6 | 19 | 7 | 8 | 10 |
| \(\frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } }\) | 0.9 | 0.4 | 10.0 | 0.4 | 0.9 | 1.6 | 8.1 | 0.9 | 0.4 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E = 10\) | B1\* | Stated or clearly implied |
| \((19-10)^2/10 = 8.1\) | B1dep | Correctly demonstrate 8.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): each number equally likely. \(H_1\): not so | B1 | Both. Oe, e.g. Uniform distribution is good model |
| \(\Sigma X^2 = 23.6\) | M1 | Add up discrepancies |
| \(\chi^2_\text{crit} = 21.67\) | A1 | Correct CV |
| Reject \(H_0\). Significant evidence that not all numbers equally likely | M1 | Correct first conclusion, ft on their \(\Sigma X^2\) but not wrong CV |
| A1 | Contextualised, not too definite |
| Answer | Marks | Guidance |
|---|---|---|
| For 3 and 7: E.g. \(p = 0.2\) or \(0.195\) | M1 | Clearly identify these two numbers |
| A1 | Any \(p\) with \(0.1 < p < 0.5\) |
## Question 6:
### Part (a):
$E = 10$ | **B1\*** | Stated or clearly implied
$(19-10)^2/10 = 8.1$ | **B1dep** | Correctly demonstrate 8.1
[2]
### Part (b):
$H_0$: each number equally likely. $H_1$: not so | **B1** | Both. Oe, e.g. Uniform distribution is good model
$\Sigma X^2 = 23.6$ | **M1** | Add up discrepancies
$\chi^2_\text{crit} = 21.67$ | **A1** | Correct CV
Reject $H_0$. Significant evidence that not all numbers equally likely | **M1** | Correct first conclusion, ft on their $\Sigma X^2$ but not wrong CV
| **A1** | Contextualised, not too definite
[5]
### Part (c):
For 3 and 7: E.g. $p = 0.2$ or $0.195$ | **M1** | Clearly identify these two numbers
| **A1** | Any $p$ with $0.1 < p < 0.5$
[2]
---6 A student believes that if you ask people to choose an integer between 1 and 10, not all integers are equally likely to be chosen. The student asks a random sample of 100 people to choose an integer between 1 and 10 inclusive. The observed frequencies $O$, together with the values of $\frac { ( O - E ) ^ { 2 } } { E }$ where $E$ is the corresponding expected frequency, are shown in the table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Integer & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
O & 7 & 8 & 20 & 8 & 7 & 6 & 19 & 7 & 8 & 10 \\
\hline
$\frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } }$ & 0.9 & 0.4 & 10.0 & 0.4 & 0.9 & 1.6 & 8.1 & 0.9 & 0.4 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Show how the value of 8.1 for integer 7 is obtained.
\item Show that there is evidence at the $1 \%$ significance level that the student's belief is correct.
The student wishes to suggest an alternative model for the probabilities associated with each integer. In this model, two of the integers have the same probability $p _ { 1 }$ of being chosen and the other eight integers each have probability $p _ { 2 }$ of being chosen.
\item Suggest which two integers should have probability $p _ { 1 }$ and suggest a possible value of $p _ { 1 }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2021 Q6 [9]}}