A-Level Mathematics October/November 2024 Q5(b): The shop also sells red apples. 60% of the red apples sold by the shop weigh more than…

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  Let $X$ be the number of red apples that weigh more than 80 grams. The distribution of $X$ is binomial, $X \sim B(160, 0.6)$.

  We can use a normal distribution to approximate this, provided $np > 5$ and $nq > 5$. $n=160$, $p=0.6$, $q=1-p=0.4$. $np = 160 \times 0.6 = 96$ $nq = 160 \times 0.4 = 64$ Since both $96 > 5$ and $64 > 5$, a normal approximation is suitable.

  The approximating normal distribution, $Y$, has mean $\mu = np$ and variance $\sigma^2 = npq$. $\mu = 96$ $\sigma^2 = 160 \times 0.6 \times 0.4 = 38.4$ So, $Y \sim N(96, 38.4)$.

  We need to find the probability that fewer than 105 apples weigh more than 80g, which is $P(X < 105)$. This is equivalent to $P(X \le 104)$.

  Applying the continuity correction, we find $P(Y < 104.5)$.

  Now we standardise the value: $Z = \frac{Y - \mu}{\sigma} = \frac{104.5 - 96}{\sqrt{38.4}}$ $Z = \frac{8.5}{6.19677...}$ $Z = 1.37168...$

  We require $P(Z < 1.3717)$. From the standard normal distribution tables, this is $\Phi(1.3717)$. $\Phi(1.3717) = 0.9149...$

  Therefore, the probability is $0.915$ (to 3 s.f.).

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  How the marks are awarded

  • B1 — For correctly calculating and stating the mean (96) and variance (38.4) of the approximating normal distribution. These values may be seen unsimplified or within the standardisation formula.
  • M1 — For substituting the calculated mean and variance into the standardisation formula, e.g., $\frac{x - 96}{\sqrt{38.4}}$.
  • M1 — For applying the correct continuity correction. The discrete probability $P(X < 105)$ or $P(X \le 104)$ becomes the continuous probability $P(Y < 104.5)$.
  • M1 — For finding the correct area under the normal curve corresponding to the calculated positive z-value. This involves looking up $\Phi(z)$ and ensuring the final probability is greater than 0.5.
  • A1 — For the correct final answer of 0.915 or a value that rounds to it, obtained from correct working.

  Common mistakes

  • Forgetting to apply a continuity correction, for example by standardising the value 105 or 104 instead of 104.5.
  • Using an incorrect continuity correction, such as 105.5, by misinterpreting 'fewer than 105' as 'up to 105' without first writing it as 'less than or equal to 104'.
  • Using the variance (38.4) instead of the standard deviation ($\sqrt{38.4}$) in the denominator of the standardisation formula.
  • Making a calculation error when finding the mean ($np$) or variance ($npq$).

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

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