| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 11 |
| 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 uniform distribution. Students must calculate expected frequencies (proportional to number of squares), compute the test statistic, and compare to critical values. The interpretation in part (ii) is routine. Slightly easier than average due to clear setup and standard procedure. |
| Spec | 5.06a Chi-squared: contingency tables |
| D | D | D | D | D | D | D |
| D | C | C | C | C | C | D |
| D | C | B | B | B | C | D |
| D | C | B | A | B | C | D |
| D | C | B | B | B | C | D |
| D | C | C | C | C | C | D |
| D | D | D | D | D | D | D |
| Region | A | B | C | D |
| Number of squares | 1 | 8 | 16 | 24 |
| Number of pins | 6 | 21 | 33 | 38 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): "No of pins \(\propto\) no of squares" is a good model | B1 | Both hypotheses stated; Equivalents acceptable, e.g. in terms of uniform dist |
| \(H_1\): "..." not a good model | B1 | |
| Expected ratios 1 : 8 : 16 : 24 | M1 | Find ratios |
| Expected frequencies 2, 16, 32, 48 | M1 | Find expected frequencies |
| Combine cells ... | M1 | Combine cells |
| ... to give 27, 33, 38; 18, 32, 48 | A1 | All correct |
| \(4.5 + 0.03125 + 2.083\) | M1* | Correct method for calculating \(\chi^2\) |
| \(\chi^2 = 6.61458\ldots\) | A1 | TS correct, in [6.61, 6.62] |
| \(> 4.605\) | B1 | Compare with 4.605 |
| Reject \(H_0\) | M1ft | Correct conclusion; FT on wrong TS only |
| Significant evidence that data does not fit model | A1ft | Acknowledge uncertainty |
| Answer | Marks | Guidance |
|---|---|---|
| The results for the inner two sectors make the largest contribution to the test statistic and they seem to be aiming at these sectors | dep A1ft | Contextualised, ft on their calculations, needs M1* |
## (i)
$H_0$: "No of pins $\propto$ no of squares" is a good model | B1 | Both hypotheses stated; Equivalents acceptable, e.g. in terms of uniform dist
$H_1$: "..." not a good model | B1 |
Expected ratios 1 : 8 : 16 : 24 | M1 | Find ratios
Expected frequencies 2, 16, 32, 48 | M1 | Find expected frequencies
Combine cells ... | M1 | Combine cells
... to give 27, 33, 38; 18, 32, 48 | A1 | All correct
$4.5 + 0.03125 + 2.083$ | M1* | Correct method for calculating $\chi^2$
$\chi^2 = 6.61458\ldots$ | A1 | TS correct, in [6.61, 6.62]
$> 4.605$ | B1 | Compare with 4.605
Reject $H_0$ | M1ft | Correct conclusion; FT on wrong TS only
Significant evidence that data does not fit model | A1ft | Acknowledge uncertainty
**Guidance:** FT on wrong TS only
## (ii)
The results for the inner two sectors make the largest contribution to the test statistic and they seem to be aiming at these sectors | dep A1ft | Contextualised, ft on their calculations, needs M1*
---7 Josh is investigating whether sticking pins into a map at random, while blindfolded, provides a random sample of regions of the map. Josh divides the map into 49 squares of equal size and asks each of 98 friends to stick a pin into the map at random, while blindfolded. He then notes the number of pins in each square. To analyse the results he groups the squares as shown in the diagram.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
D & D & D & D & D & D & D \\
\hline
D & C & C & C & C & C & D \\
\hline
D & C & B & B & B & C & D \\
\hline
D & C & B & A & B & C & D \\
\hline
D & C & B & B & B & C & D \\
\hline
D & C & C & C & C & C & D \\
\hline
D & D & D & D & D & D & D \\
\hline
\end{tabular}
\end{center}
The results are summarised in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
Region & A & B & C & D \\
\hline
Number of squares & 1 & 8 & 16 & 24 \\
\hline
Number of pins & 6 & 21 & 33 & 38 \\
\hline
\end{tabular}
\end{center}
(i) Test at the 10\% significance level whether the use of pins in this way provides a random sample of regions of the map.\\
(ii) What can be deduced from considering the different contributions to the test statistic?
\section*{OCR}
\section*{Oxford Cambridge and RSA}
\hfill \mbox{\textit{OCR FS1 AS 2017 Q7 [11]}}