Edexcel S3 — Question 7 14 marks

Exam BoardEdexcel
ModuleS3 (Statistics 3)
Marks14
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCentral limit theorem
TypeUnknown distribution, CLT applied
DifficultyStandard +0.3 This is a straightforward CLT application with clear setup: part (a) is recall, (b)-(c) identify binomial distribution parameters (routine), and (d) applies CLT to sample means with standard normal approximation. All steps are mechanical with no conceptual challenges beyond recognizing the binomial setup and applying the standard CLT formula—slightly easier than average due to explicit guidance through each step.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.05a Sample mean distribution: central limit theorem
7. (a) Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
(b) Write down the distribution of the number of sixes obtained when the ten dice are thrown.
(c) Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
(d) Find the probability that the mean number of sixes obtained is more than 1.8
AnswerMarks
(a) when a sample from any dist. is large, the dist. of the sample mean is approximately normal with same mean and variance \(\frac{\sigma^2}{n}\)B3
(b) binomial with \(n = 10\), \(p = \frac{1}{6}\)B2
(c) mean = \(np = 10 \times \frac{1}{6} = \frac{5}{3}\)A1
variance = \(npq = 10 \times \frac{1}{6} \times \frac{5}{6} = \frac{25}{18}\)M1, A1
(d) let X = no. of sixes when throw 10 dice ∴ \(X \sim B(10, \frac{1}{6})\)
AnswerMarks Guidance
∴ \(\overline{X} \sim N(\frac{5}{3}, \frac{25}{100}) = N(\frac{5}{3}, \frac{1}{12})\)M1, A2
\(P(\overline{X} > 1.8) = P(Z > \frac{1.8-\frac{5}{3}}{\sqrt{\frac{1}{12}}})\)M1
\(= P(Z > 1.13) = 1 - 0.8708 = 0.1292\)M1, A1 (14) 
**(a)** when a sample from any dist. is large, the dist. of the sample mean is approximately normal with same mean and variance $\frac{\sigma^2}{n}$ | B3 |

**(b)** binomial with $n = 10$, $p = \frac{1}{6}$ | B2 |

**(c)** mean = $np = 10 \times \frac{1}{6} = \frac{5}{3}$ | A1 |

variance = $npq = 10 \times \frac{1}{6} \times \frac{5}{6} = \frac{25}{18}$ | M1, A1 |

**(d)** let X = no. of sixes when throw 10 dice ∴ $X \sim B(10, \frac{1}{6})$

∴ $\overline{X} \sim N(\frac{5}{3}, \frac{25}{100}) = N(\frac{5}{3}, \frac{1}{12})$ | M1, A2 |

$P(\overline{X} > 1.8) = P(Z > \frac{1.8-\frac{5}{3}}{\sqrt{\frac{1}{12}}})$ | M1 |

$= P(Z > 1.13) = 1 - 0.8708 = 0.1292$ | M1, A1 | (14) |
7. (a) Briefly state the central limit theorem.

A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.\\
(b) Write down the distribution of the number of sixes obtained when the ten dice are thrown.\\
(c) Find the mean and variance of this distribution.

The student throws the ten dice 100 times, recording the number of sixes showing each time.\\
(d) Find the probability that the mean number of sixes obtained is more than 1.8

\hfill \mbox{\textit{Edexcel S3  Q7 [14]}}
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