Question

According to Environmental Canada, the yearly rainfall amount in Vancouver is normally distributed with mean 1055.4 mm and standard deviation 172 mm. a. Find the probability that the rainfall in Vancouver in any one randomly selected year is over 1000 mm? b. Find the probability that the mean rainfall in Vancouver of 10 randomly selected years is over 1000 mm? c. (5 mark) What is the probability that at least 2 of the 5 randomly selected years has rainfall over 1000 mm per year?

Answer 1

Part a)

P ( X > 1000 ) = 1 - P ( X < 1000 )
Standardizing the value

Z = ( 1000 - 1055.4 ) / 172
Z = -0.32

P ( Z > -0.32 )
P ( X > 1000 ) = 1 - P ( Z < -0.32 )
P ( X > 1000 ) = 1 - 0.3745
P ( X > 1000 ) = 0.6255

Part b)

P ( X > 1000 ) = 1 - P ( X < 1000 )
Standardizing the value


Z = -1.02

P ( Z > -1.02 )
P ( X > 1000 ) = 1 - P ( Z < -1.02 )
P ( X > 1000 ) = 1 - 0.1542
P ( X > 1000 ) = 0.8458

Part c)

P ( X > 1000 ) = 0.6255

Mean = n * P = 5 * 0.6255 = 3.1275

Standard deviation =

Using normal approximation to Binomial

P ( X >= 2 )

Using continuity correction

P ( X >= 2 ) = P ( X > 1.5 )

P ( X > 1.5 ) = 1 - P ( X < 1.5 )
Standardizing the value

Z = ( 1.5 - 3.1275 ) / 1.0822
Z = -1.5

P ( Z > -1.5 )
P ( X > 1.5 ) = 1 - P ( Z < -1.5 )
P ( X > 1.5 ) = 1 - 0.0668
P ( X > 1.5 ) = 0.9332