Question

The run time for written C-Programmes for Gauissian elimination follows a normal distribution with a mean of 36 minutes and a standard deviation of 11 minutes. For a C-Programme chosen from Gaussian elimination files at random, find the probability that the run time will be

a)        less than 51 minutes                                                                                       

b)        more than 60 minutes                                                                                    

c)        between 41 and 56 minutes                                                                           

d)        between 12 and 26 minutes 

Answer 1

Let the random variable $X$ be the run time for written C-Programmes for Gauissian elimination then, $X\sim N(\mu,\sigma^2)$

Here,

$\mu=36,\space \sigma=11$

$a)$

The probability that the run time is less than 51 minutes is given as,

$p(X\lt51)=p(Z\lt (51-\mu)/\sigma)=p(Z\lt (51-36)/11)=p(Z\lt1.36363636)$

To find the probability, we enter the following command in $R$,

> pnorm(1.36363636)

The output is,

[1] 0.913659

This output represents the probability that the run time is less than 51 minutes.

Therefore, the probability that the run time is less than 51 minutes is 0.913659.

$b)$

The probability that the run time is more than 60 minutes is given as,

$p(X\gt 60)=p(Z\gt (60-36)/11)=p(Z\gt2.18181818)=1-p(Z\lt 2.18181818)$

To find this probability, we enter the following command $R$

> 1-pnorm(2.18181818). Executing this command gives the output,

 [1] 0.01456148

This is the required probability.

Therefore, the probability that the run time is more than 60 minutes is 0.01456148.

$c)$

The probability that the run time is between 41 and 56 minutes is given as,

$p(41\lt X\lt 56)=p((41-36)/11\lt Z \lt(56-36)/11)=p(0.455\lt Z\lt 1.818)=\phi(1.818)-\phi(0.455)= 0.9654818- 0.6752819= 0.2902$

Therefore, the probability that the run time is between 41 and 56 minutes is 0.2902.

$d)$

The probability that the run time is  between 12 and 26 minutes is given as,

$p(12\lt X\lt 26)=p((12-36)/11\lt Z\lt (26-36)/11)=P(-2.18\lt Z\lt -0.91)=\phi(-0.91)-\phi(-2.18)= 0.1816511- 0.01456148= 0.1670896$

Therefore, the probability that the run time is between 12 and 26 minutes is 0.1670896.