| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Uniform |
| Difficulty | Standard +0.3 This is a straightforward chi-squared goodness of fit test with equal expected frequencies (uniform distribution). Students need to state hypotheses, calculate expected frequencies (16 each), compute the test statistic with 4 degrees of freedom, and compare to critical value. It's slightly easier than average because the uniform distribution makes calculations simple, requires no parameter estimation, and follows a standard template that S3 students practice extensively. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Last Digit | 1 | 3 | 5 | 7 | 9 |
| Frequency | 16 | 20 | 14 | 17 | 13 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_1\): discrete uniform is not a suitable model | B1 | |
| exp. freqs = 80 ÷ 5 = 16 | M1, A1 | |
| O | E | (O - E) |
| 16 | 16 | 0 |
| 20 | 16 | 4 |
| 14 | 16 | −2 |
| 17 | 16 | 1 |
| 13 | 16 | −3 |
| \(\sum \frac{(O-E)^2}{E} = 1.875\) | M1, A2 | |
| \(\nu = 5 - 1 = 4\), \(\chi^2_{crit}(10\%) = 7.779\) | M1, A1 | |
| 1.875 < 7.779 ∴ do not reject \(H_0\) discrete uniform is a suitable model supporting psychologist's theory | A1 | (9) |
$H_0$: discrete uniform is a suitable model
$H_1$: discrete uniform is not a suitable model | B1 |
exp. freqs = 80 ÷ 5 = 16 | M1, A1 |
| O | E | (O - E) | $\frac{(O-E)^2}{E}$ |
|---|---|---------|------------|
| 16 | 16 | 0 | 0 |
| 20 | 16 | 4 | 1 |
| 14 | 16 | −2 | 0.25 |
| 17 | 16 | 1 | 0.0625 |
| 13 | 16 | −3 | 0.5625 |
$\sum \frac{(O-E)^2}{E} = 1.875$ | M1, A2 |
$\nu = 5 - 1 = 4$, $\chi^2_{crit}(10\%) = 7.779$ | M1, A1 |
1.875 < 7.779 ∴ do not reject $H_0$ discrete uniform is a suitable model supporting psychologist's theory | A1 | (9) |
---\begin{enumerate}
\item A psychologist is investigating the numbers people choose when asked to pick a number at random in a given interval. He finds that when asked to pick a number between 0 and 100 people are less likely to pick certain numbers, such as multiples of ten. He believes, however that if people are asked to pick an odd number between 0 and 100 they are equally likely to pick a number ending in any of the digits $1,3,5,7$ or 9 .
\end{enumerate}
To test this theory he asks 80 people to pick an odd number between 0 and 100 and records the last digit of the numbers chosen. His results are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Last Digit & 1 & 3 & 5 & 7 & 9 \\
\hline
Frequency & 16 & 20 & 14 & 17 & 13 \\
\hline
\end{tabular}
\end{center}
Stating your hypotheses clearly and using a 10\% level of significance test the psychologist's theory.\\
(9 marks)\\
\hfill \mbox{\textit{Edexcel S3 Q2 [9]}}