| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared distribution theory and properties |
| Difficulty | Standard +0.8 This S4 Further Maths question requires manipulating moment generating functions to find moments, recognizing that sums of chi-squared variables follow chi-squared distributions, and applying normal approximations. While the techniques are standard for S4, the MGF manipulation and distribution identification require solid theoretical understanding beyond routine calculation, placing it moderately above average difficulty. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| EITHER: (a) \(M'(t) = n(1-2t)^{-\frac{1}{2}n-1}\) | M1 A1 | Correct form for M1 |
| \(E(Y) = M'(0) = n\) | A1 | |
| \(M''(t) = n(n+2)(1-2t)^{-\frac{1}{2}n-2}\) | M1 | ft similar \(M'(t)\) |
| \(\text{Var}(Y) = n(n+2) - n^2 = 2n\) | A1 [5] | \(M''(0) - (M'(0))^2\) |
| OR: \(M(t) = 1 + nt + \frac{1}{2}n(n+2)t^2\) | M1A1A1 | |
| \(E(Y) = n\) | A1 | |
| \(\text{Var}(Y) = n(n+2) - n^2 = 2n\) | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\text{MGF} = (1-2t)^{-30}\) | B1 | From \([(1-2t)^{-1/2}]^{60}\) |
| \(\chi^2\) distribution with 60 d.f. | B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(S) = 60\), \(\text{Var}(S) = 120\) | B1ft | From (i) |
| Using CLT, Probability \(= 1 - \Phi(10/\sqrt{120})\) | M1 | Correct tail: allow cc |
| \(= 0.181\) | A1 [3] |
## Question 4:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| EITHER: (a) $M'(t) = n(1-2t)^{-\frac{1}{2}n-1}$ | M1 A1 | Correct form for M1 |
| $E(Y) = M'(0) = n$ | A1 | |
| $M''(t) = n(n+2)(1-2t)^{-\frac{1}{2}n-2}$ | M1 | ft similar $M'(t)$ |
| $\text{Var}(Y) = n(n+2) - n^2 = 2n$ | A1 **[5]** | $M''(0) - (M'(0))^2$ |
| OR: $M(t) = 1 + nt + \frac{1}{2}n(n+2)t^2$ | M1A1A1 | |
| $E(Y) = n$ | A1 | |
| $\text{Var}(Y) = n(n+2) - n^2 = 2n$ | A1 **[5]** | |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{MGF} = (1-2t)^{-30}$ | B1 | From $[(1-2t)^{-1/2}]^{60}$ |
| $\chi^2$ distribution with 60 d.f. | B1 **[2]** | |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(S) = 60$, $\text{Var}(S) = 120$ | B1ft | From (i) |
| Using CLT, Probability $= 1 - \Phi(10/\sqrt{120})$ | M1 | Correct tail: allow cc |
| $= 0.181$ | A1 **[3]** | |
---4 The moment generating function of a continuous random variable $Y$, which has a $\chi ^ { 2 }$ distribution with $n$ degrees of freedom, is $( 1 - 2 t ) ^ { - \frac { 1 } { 2 } n }$, where $0 \leqslant t < \frac { 1 } { 2 }$.\\
(i) Find $\mathrm { E } ( Y )$ and $\operatorname { Var } ( Y )$.
For the case $n = 1$, the sum of 60 independent observations of $Y$ is denoted by $S$.\\
(ii) Write down the moment generating function of $S$ and hence identify the distribution of $S$.\\
(iii) Use a normal approximation to estimate $\mathrm { P } ( S \geqslant 70 )$.
\hfill \mbox{\textit{OCR S4 2010 Q4 [10]}}