Question

On a night drive, a driver’s response time to any animal crossing the highway is normally distributed with a mean of 0.4 seconds and a standard deviation of 0.05 seconds. draw standard normal graphs and shade the regions clearly.

1. What is the probability that a response takes more than 0.5 seconds?

   

2. What is the probability that a response takes between 0.4 and 0.5 seconds?

  

3. What response time is exceeded 90% of the time?

Answer 1

Solution :

P(x > 0.5) = 1 - P(x < 0.5)

= 1 - P[(x - ) / < (0.5 - 0.4) / 0.05)

= 1 - P(z < 2)

= 1 - 0.9772

= 0.0228

probability = 0.0228

P(0.4 < x < 0.5) = P[(0.4 - 0.4)/ 0.05) < (x - ) /  < (0.5 - 0.4) / 0.05) ]

= P(0 < z < 2)

= P(z < 2) - P(z < 0)

= 0.9772 - 0.5

= 0.4772

probability = 0.4772

Using standard normal table ,

P(Z > z) = 90%

1 - P(Z < z) = 0.9

P(Z < z) = 1 - 0.9

P(Z < -1.28) = 0.1

z = -1.28

Using z-score formula,

x = z * +

x = -1.28 * 0.05 + 0.4 = 0.34

response time is exceeded 90% of the time is 0.34