Question

the number of miles that Anita's motorbike will be travel on one gallon of petrol may be modelled by a normal distribution with mean 135 and standard deviation 12.a) given that Anita starts journey with one gallon of petrol in her motorbike's tank, find the probability that, without refeulling,she can travel.i) more than 111 miles,ii) between 141 and 150 miles.b) find the longest journey that Anita can undertake,if she is to have a probability of that at least 0.9 of completing it on one gallon of petrol

Answer 1

X=Number of miles that Anita's motorbike will be travel on one gallon of petrol

Standard normal distribution is obtain by using,

$z=\frac{x-\bar{x}}{\sigma}=\frac{x-135}{12}\\$

12)a)

i)

$P(X>111)=P(Z>\frac{111-135}{12})\\ P(X>111)=P(Z>-2)\\ P(X>111)=0.9772\\$

Probability that, without refeulling,she can travel more than 111 miles=0.9772

ii)

$P(141<X<150)=P(\frac{141-135}{12}<Z<\frac{150-135}{12})\\ P(141<X<150)=P(0.5<Z<1.25)\\ P(141<X<150)=P(0<Z<1.25)\\ \hspace{12 em}-P(0<Z<0.5)\\ P(141<X<150)=0.3944-0.1915\\ P(141<X<150)=0.2029\\$

 Probability that, without refeulling,she can travel between 141 and 150 miles=0.2029

b)task is to find z such that,

$P(Z<z)=0.9\\ z=1.282\\ z=\frac{x-135}{12}\\ 1.282=\frac{x-135}{12}\\ x=150.384$

longest journey that Anita can undertake,if she is to have a probability of that at least 0.9 of completing it on one gallon of petrol is 150.384 miles