| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared test of independence |
| Difficulty | Challenging +1.2 This is a chi-squared test with algebraic frequencies requiring calculation of expected values with variables, then a standard hypothesis test. Part (a) requires careful algebraic manipulation of the expected value formula (row total × column total / grand total) with multiple variables, which is more demanding than numerical chi-squared questions. Part (b) is routine once the test statistic is given. The algebraic component elevates this above a standard chi-squared question (which would be ~0.0) but it's still a straightforward application of learned techniques without requiring novel insight. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Main type of investment | |||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Bonds | Cash | Stocks | |
| \multirow{2}{*}{Age} | \(25 - 44\) | \(a\) | \(b - e\) | \(e\) |
| \cline { 2 - 5 } | \(45 - 75\) | \(c\) | \(d - 59\) | 59 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E = \frac{(c+d)(a+c)}{a+b+c+d}\) | B1 | For correct expected value |
| \(O - E = c - \text{"}\frac{(c+d)(a+c)}{a+b+c+d}\text{"}\) | M1 | For finding \(c\) – their expected value |
| \(O - E = \frac{ca+cb+c^2+cd-ac-c^2-ad-dc}{a+b+c+d}\) | dM1 | Dependent on previous method being awarded. For correctly gaining a single fraction |
| \(O - E = \frac{cb-ad}{a+b+c+d}\) | A1 | Correct answer only |
| (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): There is no association between the age of a person and the main type of investment they have. \(H_1\): There is an association between the age of a person and the main type of investment they have. | B1 | For correct hypotheses with at least one in context. Allow *independent* and *not independent*. Do not accept *correlation* |
| Degrees of freedom \(= (3-1)(2-1) = 2\); \(\chi^2_{2,0.05} = 5.991\) | M1 | For using degrees of freedom to set up \(\chi^2\) model critical value, implied by CV 5.991 or better |
| Reject \(H_0\). There is evidence that there is an association between the age of a person and the main type of investment they have. | A1 | Correct conclusion including the words age and investment. Do not allow contradicting statements. Do not award if hypotheses are the wrong way round or there are no hypotheses. |
| (3) |
# Question 4:
## Part 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E = \frac{(c+d)(a+c)}{a+b+c+d}$ | B1 | For correct expected value |
| $O - E = c - \text{"}\frac{(c+d)(a+c)}{a+b+c+d}\text{"}$ | M1 | For finding $c$ – their expected value |
| $O - E = \frac{ca+cb+c^2+cd-ac-c^2-ad-dc}{a+b+c+d}$ | dM1 | Dependent on previous method being awarded. For correctly gaining a single fraction |
| $O - E = \frac{cb-ad}{a+b+c+d}$ | A1 | Correct answer only |
| | **(4)** | |
## Part 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: There is no association between the age of a person and the main type of investment they have. $H_1$: There is an association between the age of a person and the main type of investment they have. | B1 | For correct hypotheses with at least one in context. Allow *independent* and *not independent*. Do not accept *correlation* |
| Degrees of freedom $= (3-1)(2-1) = 2$; $\chi^2_{2,0.05} = 5.991$ | M1 | For using degrees of freedom to set up $\chi^2$ model critical value, implied by CV 5.991 or better |
| Reject $H_0$. There is evidence that there is an association between the age of a person and the main type of investment they have. | A1 | Correct conclusion including the words **age** and **investment**. Do not allow contradicting statements. Do not award if hypotheses are the wrong way round or there are no hypotheses. |
| | **(3)** | |\begin{enumerate}
\item Charlie carried out a survey on the main type of investment people have.
\end{enumerate}
The contingency table below shows the results of a survey of a random sample of people.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{c|}{Main type of investment} \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & Bonds & Cash & Stocks \\
\hline
\multirow{2}{*}{Age} & $25 - 44$ & $a$ & $b - e$ & $e$ \\
\cline { 2 - 5 }
& $45 - 75$ & $c$ & $d - 59$ & 59 \\
\hline
\end{tabular}
\end{center}
(a) Find an expression, in terms of $a , b , c$ and $d$, for the difference between the observed and the expected value $( O - E )$ for the group whose main type of investment is Bonds and are aged 45-75\\
Express your answer as a single fraction in its simplest form.
Given that $\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62$ for this information,\\
(b) test, at the $5 \%$ level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.
\hfill \mbox{\textit{Edexcel FS1 AS 2021 Q4 [7]}}