A-Level Mathematics May/June 2024 Q5(c): Use a suitable approximating distribution to find the probability that the number of sa…

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  Let $X$ be the number of sales in a 140-hour period. The sales occur at a constant average rate, so $X$ follows a Poisson distribution. The mean rate, $\lambda = 140 \times 1.2 = 168$. So, $X \sim \text{Po}(168)$.

  Since $\lambda = 168 > 15$, a Normal distribution is a suitable approximation. Let $Y$ be the approximating Normal distribution. $Y \sim \text{N}(\mu, \sigma^2)$, where $\mu = \lambda = 168$ and $\sigma^2 = \lambda = 168$. So, $Y \sim \text{N}(168, 168)$.

  We need to find the probability that the number of sales is less than 150, which is $P(X < 150)$. This is equivalent to $P(X \le 149)$. Applying the continuity correction, we find $P(Y < 149.5)$.

  Standardise the value: $Z = \frac{Y - \mu}{\sigma} = \frac{149.5 - 168}{\sqrt{168}}$ $Z = \frac{-18.5}{12.96148...} = -1.4273...$

  We require $P(Z < -1.4273)$. From the standard normal distribution tables: $\Phi(-1.4273) = 1 - \Phi(1.4273)$ $= 1 - 0.9232$ $= 0.0768$

  The probability is 0.0768 (3 s.f.).

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  How the marks are awarded

  • B1 — Stating or implying the use of the Normal distribution N(168, 168) as an approximation to the Poisson distribution.
  • M1 — Applying the standardisation formula with a continuity correction. The calculation (149.5 - 168) / √168 earns this mark.
  • M1 — Finding the correct tail probability for the calculated z-score. This involves recognising that for a negative z-value, P(Z < z) = 1 - P(Z < -z), i.e., calculating 1 - Φ(1.427).
  • A1 — Obtaining the final correct answer of 0.0768 or 0.0767, correctly rounded to 3 significant figures.

  Common mistakes

  • Forgetting to apply the continuity correction, i.e., using 150 instead of 149.5 in the standardisation formula.
  • Applying the continuity correction incorrectly, for example using 150.5 for a 'less than' probability.
  • Using the variance (168) instead of the standard deviation (√168) in the denominator of the standardisation formula.
  • Incorrectly reading the standard normal distribution table, for example finding Φ(1.427) but forgetting to subtract it from 1.

  Examiner tip: When approximating a discrete distribution with a continuous one, always apply a continuity correction by adding or subtracting 0.5 from the discrete value before standardising.

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