S1: The Normal distributionEdexcel International A Level Maths: Topic test

1. 1
   The lifetime, $L$ hours, of a type of battery is Normally distributed with mean 120 and standard deviation 15.

   (a)

   Find $\mathrm{P}(L>135)$.

   [1 mark]

   • A$0.8413$
   • B$0.1587$
   • C$0.3413$
   • D$0.0228$

   (b)

   A battery lasts for 97.5 hours. Find its standardised value $z$.

   [1 mark]

   • A$1.5$
   • B$-22.5$
   • C$-1.5$
   • D$-0.1$

   (c)

   Find $\mathrm{P}(105<L<150)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The masses of ripe mangoes from an orchard are Normally distributed with mean 340 g and standard deviation 25 g.

   (a)

   A mango has mass 390 g. Find its standardised value $z$.

   [1 mark]

   • A$50$
   • B$0.08$
   • C$-2$
   • D$2$

   (b)

   Find the probability that a mango has mass less than 352.5 g.

   [1 mark]

   • A$0.6915$
   • B$0.3085$
   • C$0.8413$
   • D$0.5000$

   (c)

   Find the mass that is exceeded by 10% of the mangoes, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The volume, $V$ ml, of juice in a carton filled by a machine is Normally distributed with mean $\mu$ and standard deviation $\sigma$.

   (a)

   The machine is set so that $\sigma=4$. Given that 5% of cartons contain less than 245 ml, find the value of $\mu$, giving your answer to 1 decimal place.

   [3 marks]

   (b)

   The machine is reset so that $\mu=252$. Given that 1% of cartons contain less than 240 ml, find the value of $\sigma$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The times, $T$ minutes, taken by students to complete a puzzle are Normally distributed with mean 18 and standard deviation 4.

   (a)

   (i) Find $\mathrm{P}(14<T<24)$.
   (ii) Find the time that is exceeded by 5% of the students, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   A second group of students has times, $U$ minutes, that are Normally distributed with mean $\mu$ and standard deviation $\sigma$. Of this group, 10% take less than 10 minutes and 20% take more than 20 minutes. Find $\mu$ and $\sigma$, giving each answer to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The wingspan, $W$ cm, of a species of butterfly is Normally distributed with $W\sim\mathrm{N}(7.2,\,0.64)$.

   (a)

   State the standard deviation of $W$.

   [1 mark]

   • A$0.64$
   • B$0.8$
   • C$0.32$
   • D$0.08$

   (b)

   Find $\mathrm{P}(W>8.8)$.

   [1 mark]

   • A$0.9772$
   • B$0.0062$
   • C$0.0548$
   • D$0.0228$

   (c)

   Find $\mathrm{P}(6.4<W<8.0)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The scores on a driving theory test are Normally distributed with mean 41 and standard deviation 5.

   (a)

   Find the probability that a candidate scores more than 47.

   [1 mark]

   • A$0.1151$
   • B$0.8849$
   • C$0.2743$
   • D$0.7257$

   (b)

   Find the upper quartile of the scores.

   [1 mark]

   • A$37.6$
   • B$42.3$
   • C$44.4$
   • D$46.0$

   (c)

   Find $\mathrm{P}(38<X<47)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The resistance, $R$ ohms, of resistors made in a factory is Normally distributed with mean 100 and standard deviation 2.5. A resistor is rejected if its resistance is less than 96 or greater than 105.

   (a)

   Find the probability that a resistor is rejected.

   [3 marks]

   (b)

   The mean stays at 100 ohms but the standard deviation is changed to $\sigma$ so that only 1% of resistors have a resistance greater than 105 ohms. Find $\sigma$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A machine cuts metal rods. The length, $X$ mm, of a rod is Normally distributed with mean 250 and standard deviation 1.25. A rod is acceptable if its length is between 247.5 mm and 252.5 mm.

   (a)

   (i) Find the probability that a rod is acceptable.
   (ii) The machine is adjusted so that the mean is still 250 mm but the standard deviation is reduced to $\sigma$, and 99% of rods are acceptable. Find $\sigma$, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   A second machine produces rods whose lengths are Normally distributed with mean $\mu$ and standard deviation $\sigma$. Of these rods, 5% are shorter than 247.5 mm and 15% are longer than 252.5 mm. Find $\mu$ and $\sigma$, giving $\sigma$ to 3 significant figures and $\mu$ to 1 decimal place.

   [6 marks]

   Total for question 8: 12 marks

End of questions