Statistics indicate that the rainfall in Ithaca New York has

Statistics indicate that the rainfall in Ithaca, New York has mean Mu = 35 4 \"with standard deviation sigma = 4.2\". Assume that a Normal model applies. During what percentage of years does Ithaca is the mean rain fall more than 40\" of rain About how much rain falls in the driest 20% of all years (What is the cut-off value for the lowest 20% of rainfall) A Cornell University student is in Ithaca for 4 years Let x bar represent the mean amount of rain that falls in those 4 years. Describe the Sampling Distribution of the Sample Mean for all the different values of the sample mean amount of rain that falls in four years. What is the probability that those 4 years average less than 30\"of rain P(x bar less than or equal to 30)

Solution

Mean ( u ) =35.4
Standard Deviation ( sd )=4.2
Normal Distribution = Z= X- u / sd ~ N(0,1)                  

a)
P(X > 40) = (40-35.4)/4.2
= 4.6/4.2 = 1.0952
= P ( Z >1.095) From Standard Normal Table
= 0.1367                  

13.67 ~ 14% of years does ithaca is the mean rainfall
more than 40

b)
P ( Z < x ) = 0.2
Value of z to the cumulative probability of 0.2 from normal table is -0.842
P( x-u/s.d < x - 35.4/4.2 ) = 0.2
That is, ( x - 35.4/4.2 ) = -0.84
--> x = -0.84 * 4.2 + 35.4 = 31.8636                  

It is about 31.86

d)
Mean ( u ) =35.4
Standard Deviation ( sd )=4.2/ Sqrt ( 4 ) = 2.1
Number ( n ) = 4
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  

P(X > 30) = (30-35.4)/4.2/ Sqrt ( 4 )
= -5.4/2.1= -2.5714
= P ( Z >-2.5714) From Standard Normal Table
= 0.9949                  
P(X < = 30) = (1 - P(X > 30)
= 1 - 0.9949 = 0.0051