| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| 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 expected ratios. Parts (a)-(c) are routine application of the standard procedure (calculate expected frequencies from 9:3:3:1, compute test statistic, compare to critical value). Part (d) requires minor interpretation of residuals but is still guided. Slightly above average difficulty due to the multi-part nature and part (d) requiring some thought, but all techniques are standard for Further Statistics. |
| Spec | 5.06a Chi-squared: contingency tables |
| Type | \(a\) | \(b\) | \(c\) | \(d\) |
| Frequency | 51 | 33 | 30 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): population frequencies in ratio \(9:3:3:1\); \(H_1\): population frequencies not in ratio \(9:3:3:1\) | B1 [1] | "Population" can be implied, but *not* "observed frequencies are in ratio .." |
| Answer | Marks | Guidance |
|---|---|---|
| Expected frequencies \(67.5,\ 22.5,\ 22.5,\ 7.5\) | B1 | At least 2 correct |
| \(\sum\dfrac{(O-E)^2}{E} = 4.033 + 4.9 + 2.5 + 0.3\) | M1 | Correct method for \(X^2\) |
| \(= 11.73\) | A1 | Awrt 11.7 BC (working not needed) |
| \(11.73 > 11.34\) | A1 | Compare \(X^2\) value to CV 11.34 |
| Reject \(H_0\). There is significant evidence that the results are not consistent with theory | M1ft A1ft [6] | ft on TS only. Needs valid TS; Conclusion, contextualised, acknowledge uncertainty. *Not* FT on hypotheses reversed |
| Answer | Marks | Guidance |
|---|---|---|
| \(b\) as it makes the largest contribution to \(X^2\) | B1 [1] | Must refer to test statistic, not "difference" |
| Answer | Marks | Guidance |
|---|---|---|
| Increase \(b\) (from 3) and *either* decrease \(a\) (from 9) *or* increase \(c\) (from 3), e.g. \(7:5:3:1\) or \(9:5:5:1\) | B1 [1] | E.g. \(7:4:3:1\), \(8:4:3:1\) or \(9:4:4:1\); Allow description if fully correct |
# Question 7:
## Part (a)
$H_0$: population frequencies in ratio $9:3:3:1$; $H_1$: population frequencies not in ratio $9:3:3:1$ | **B1 [1]** | "Population" can be implied, but *not* "observed frequencies are in ratio .." | Allow "data consistent with model"
## Part (b)
Expected frequencies $67.5,\ 22.5,\ 22.5,\ 7.5$ | **B1** | At least 2 correct | Allow rounded if seen correct
$\sum\dfrac{(O-E)^2}{E} = 4.033 + 4.9 + 2.5 + 0.3$ | **M1** | Correct method for $X^2$
$= 11.73$ | **A1** | Awrt 11.7 **BC** (working not needed)
$11.73 > 11.34$ | **A1** | Compare $X^2$ value to CV 11.34 | Must see (any) value of $X^2$
Reject $H_0$. There is significant evidence that the results are not consistent with theory | **M1ft A1ft [6]** | ft on TS only. Needs valid TS; Conclusion, contextualised, acknowledge uncertainty. *Not* FT on hypotheses reversed | *Not* "results are not consistent". FT on "do not reject" needs double negative
## Part (c)
$b$ as it makes the largest contribution to $X^2$ | **B1 [1]** | Must refer to test statistic, not "difference" | Or "33" or other clear identification
## Part (d)
Increase $b$ (from 3) and *either* decrease $a$ (from 9) *or* increase $c$ (from 3), e.g. $7:5:3:1$ or $9:5:5:1$ | **B1 [1]** | E.g. $7:4:3:1$, $8:4:3:1$ or $9:4:4:1$; Allow description if fully correct | Inconsistent to increase $c$ but not $b$
---7 In a standard model from genetic theory, the ratios of types $a , b , c$ and $d$ of a characteristic from a genetic cross are predicted to be 9:3:3:1. Andrei collects 120 specimens from such a cross, and the numbers corresponding to each type of the characteristic are given in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
Type & $a$ & $b$ & $c$ & $d$ \\
\hline
Frequency & 51 & 33 & 30 & 6 \\
\hline
\end{tabular}
\end{center}
Andrei tests, at the 1\% significance level, whether the observed frequencies are consistent with the standard model.
\begin{enumerate}[label=(\alph*)]
\item State appropriate hypotheses for the test.
\item Carry out the test.
\item State with a reason which one of the frequencies is least consistent with the standard model.
\item Suggest a different, improved model by changing exactly two of the ratio values.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2019 Q7 [9]}}