| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Given ratios |
| Difficulty | Standard +0.3 This is a straightforward chi-squared goodness of fit test with given probabilities and frequencies. Students need to calculate expected frequencies (multiply probabilities by 80), compute the test statistic using the standard formula, find degrees of freedom (4), compare to critical value, and state a conclusion. It's slightly easier than average because the probabilities are given explicitly, no cells need combining, and it follows a standard template with no conceptual complications. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Score | 1 | 2 | 3 | 4 | 6 |
| Probability | \(\frac { 3 } { 10 }\) | \(\frac { 1 } { 10 }\) | \(\frac { 1 } { 10 }\) | \(\frac { 2 } { 5 }\) | \(\frac { 1 } { 10 }\) |
| Score | 1 | 2 | 3 | 4 | 6 |
| Frequency | 15 | 4 | 12 | 41 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | \(H_0\): Spinner is working as designed (o.e.); \(H_1\): Spinner is not working as designed (o.e.) | B1 |
| (b) | \(\sum \frac{(O_i - E_i)^2}{E_i} = 3.375 + 2 + 2 + 2.53125 + 0 = 9.90625\); or \(\sum \frac{O_i^2}{E_i} - N = 9.375 + 2 + 18 + 52.53125 + 8 - 80 = 9.90625\) | M1, A1 |
| (c) | \(\nu = 5 - 1 = 4\) so \(\chi_c^2(10\%)\) cv = 7.779 or better. Result is significant so there is evidence that the spinner is not operating as designed | B1, A1cso |
| (7 marks) |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | $H_0$: Spinner is working as designed (o.e.); $H_1$: Spinner is not working as designed (o.e.) | B1 | 1st B1 for both hypotheses given in suitable context |
| (b) | $\sum \frac{(O_i - E_i)^2}{E_i} = 3.375 + 2 + 2 + 2.53125 + 0 = 9.90625$; or $\sum \frac{O_i^2}{E_i} - N = 9.375 + 2 + 18 + 52.53125 + 8 - 80 = 9.90625$ | M1, A1 | 1st M1 for using the model to find at least 2 correct expected frequencies. 2nd A1 for all correct $E_i$. 2nd M1 for attempt to find test statistic (at least two correct expressions, fractions or decimals). 2nd A1 for a correct test statistic (awrt 9.91) [ accept $\frac{32}{32}$ ] |
| (c) | $\nu = 5 - 1 = 4$ so $\chi_c^2(10\%)$ cv = 7.779 or better. Result is significant so there is evidence that the spinner is not operating as designed | B1, A1cso | B1 for correct critical value (allow 7.78). A1cso dep on all previous marks for a correct conclusion in context (can be in terms of model or spinner's design). Must mention spinner and scores or design. Accept "spinner is not accurate" |
| | | **(7 marks)** | |\begin{enumerate}
\item A spinner used for a game is designed to give scores with the following probabilities
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Score & 1 & 2 & 3 & 4 & 6 \\
\hline
Probability & $\frac { 3 } { 10 }$ & $\frac { 1 } { 10 }$ & $\frac { 1 } { 10 }$ & $\frac { 2 } { 5 }$ & $\frac { 1 } { 10 }$ \\
\hline
\end{tabular}
\end{center}
The spinner is spun 80 times and the results are as follows
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Score & 1 & 2 & 3 & 4 & 6 \\
\hline
Frequency & 15 & 4 & 12 & 41 & 8 \\
\hline
\end{tabular}
\end{center}
Test, at the $10 \%$ level of significance, whether or not the spinner is giving scores as it is designed to do. Show your working and state your hypotheses clearly.
\hfill \mbox{\textit{Edexcel FS1 AS 2019 Q2 [7]}}