| Date | May 2016 | Marks available | 4 | Reference code | 16M.1.hl.TZ0.2 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Question number | 2 | Adapted from | N/A |
Question
The lifetime, in years, of a randomly chosen basic vacuum cleaner is assumed to be modelled by the normal distribution \(B \sim {\text{N}}(14,{\text{ }}{3^2})\).
The lifetime, in years, of a randomly chosen robust vacuum cleaner is assumed to be modelled by the normal distribution \(R \sim {\text{N}}(20,{\text{ }}{4^2})\).
Find \({\text{P}}\left( {B > {\text{E}}(B) + \frac{1}{2}\sqrt {{\text{Var}}(B)} } \right)\).
Find the probability that the total lifetime of 7 randomly chosen basic vacuum cleaners is less than 100 years.
Find the probability that the total lifetime of 5 randomly chosen robust vacuum cleaners is greater than the total lifetime of 7 randomly chosen basic vacuum cleaners.
Markscheme
\({\text{P}}(B > 15.5){\text{ }}\left( { = {\text{P}}(Z > 0.5)} \right)\) (M1)
\( = (1 - 0.69146) = 0.309\) A1
[2 marks]
consider \(V = {B_1} + {B_2} + {B_3} + {B_4} + {B_5} + {B_6} + {B_7}\) (M1)
\({\text{E}}(V) = 98\) (A1)
\({\text{Var}}(V) = 63\) or equivalent (A1)
Note: No need to state \(V\) is normal.
\({\text{P}}(V < 100) = \left( {{\text{P}}\left( {Z < \frac{2}{{\sqrt {63} }} = 0.251976 \ldots } \right)} \right) = 0.599\) A1
[4 marks]
consider \(W = {R_1} + {R_2} + {R_3} + {R_4} + {R_5} - ({B_1} + {B_2} + {B_3} + {B_4} + {B_5} + {B_6} + {B_7})\) (M1)
\({\text{E}}(W) = 2\) (A1)
\({\text{Var}}(W) = 80 + 63 = 143\) (A1)
\({\text{P}}(W > 0) = \left( {{\text{P}}\left( {Z < \frac{2}{{\sqrt {143} }}} \right)} \right)\) (M1)
\( = 0.566\) A1
[5 marks]
Examiners report
This was one of the more successful questions on the paper with many wholly correct answers seen. Only a very small number failed to complete part (a) successfully. There were also many fully correct answers to part (b). Part (c) caused a problem for some candidates where in most of those cases they failed to calculate the variance correctly.
This was one of the more successful questions on the paper with many wholly correct answers seen. Only a very small number failed to complete part (a) successfully. There were also many fully correct answers to part (b). Part (c) caused a problem for some candidates where in most of those cases they failed to calculate the variance correctly.
This was one of the more successful questions on the paper with many wholly correct answers seen. Only a very small number failed to complete part (a) successfully. There were also many fully correct answers to part (b). Part (c) caused a problem for some candidates where in most of those cases they failed to calculate the variance correctly.
