The Normal distributionAQA A-Level Maths: Subtopic test

1. 1
   The mass of eggs from a farm is modelled by the random variable $X\sim N(60,4^2)$, where $X$ is measured in grams.

   (a)

   Find the probability that an egg has a mass greater than $64$ g.

   [1 mark]

   • A$0.8413$
   • B$0.1587$
   • C$0.3085$
   • D$0.0228$

   (b)

   The graph of the probability density function of $X$ has two points of inflection. Their $x$-coordinates are

   [1 mark]

   • A$52$ and $68$
   • B$56$ and $60$
   • C$44$ and $76$
   • D$56$ and $64$

   (c)

   Eggs with a mass between $56$ g and $63$ g are classed as medium. Find the probability that an egg chosen at random is medium.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The time $T$ minutes that students take to complete a puzzle is modelled by the Normal distribution with mean $25$ and standard deviation $6$.

   (a)

   Find the probability that a student takes less than $20$ minutes.

   [1 mark]

   • A$0.2023$
   • B$0.7977$
   • C$0.4448$
   • D$0.2977$

   (b)

   $90\%$ of students complete the puzzle in less than $t$ minutes. Find $t$.

   [1 mark]

   • A$26.3$
   • B$30.4$
   • C$32.7$
   • D$17.3$

   (c)

   Find the interquartile range of $T$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A nursery models the height $Y$ cm of its two-year-old saplings as $Y\sim N(\mu,\sigma^2)$. It is found that $20\%$ of saplings are shorter than $30$ cm and $10\%$ of saplings are taller than $45$ cm.

   (a)

   Find the values of $\mu$ and $\sigma$.

   [3 marks]

   (b)

   Find the probability that a sapling is taller than $40$ cm. Five saplings are chosen at random and independently. Find the probability that exactly two of them are taller than $40$ cm.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A machine fills bags of flour. The mass $X$ g of a bag is modelled by $X\sim N(500,8^2)$. The bags are labelled as containing at least $485$ g.

   (a)

   The manufacturer requires that no more than $1\%$ of bags contain less than $485$ g.
   (i) Find the probability that a bag contains less than $485$ g.
   (ii) A batch has $2000$ bags. Find the expected number of bags in the batch with less than $485$ g, and state, with a reason, whether the requirement is met.
   (iii) The mean is adjusted to $\mu$ g with the standard deviation unchanged. Find the least integer value of $\mu$ for which the requirement is met.

   [6 marks]

   (b)

   A second machine fills bags with mass $Y$ g, where $Y\sim N(500,5^2)$.
   (i) Write down the $x$-coordinates of the points of inflection of the probability density function of $X$.
   (ii) Find $P(492<X<508)$ and $P(492<Y<508)$.
   (iii) Use your answers to explain which machine is more consistent.

   [6 marks]

   Total for question 4: 12 marks

End of questions