Answer to Question #268311 in Statistics and Probability for ggggg

Question

Fifty matched pairs of magnitude/depth measurements randomly selected from 10,594

earthquakes recorded in one year from a location in California, USA. Assume that Richter

scale magnitudes of earthquakes are normally distributed with a mean of 1.184 and a

standard deviation of 0.587.

a) Earthquakes with magnitudes less than 2.00 are considered “micro-earthquakes”

that are not felt. What percentage of earthquakes fall into this category?

b) Earthquakes above 4.0 will cause indoor items to shake. What percentage of

earthquakes fall into this category?

c) Find the 90th percentile. Will all earthquakes above the 90th percentile cause

indoor items to shake?

Answer 1

Let X be a random variable represents the magnitude of an earthquake, then X ~ $N(1.184,0.587^2) = 1.184 + 0.587N(0,1)$

(a) $P(X<2)=P(1.184+0.587N(0,1)<2)=P(N(0,1)<1.39)=0.91774$

So, there is 0.91774*100%=91.8% of earthquakes are microearthquakes

(b) $P(X>4)=P(1.184+0.587N(0,1)>4)=P(N(0,1)>4.8)=0.0000008$

So, there is 0.0000008*100%=0.00008% of earthquakes are cause indoors items to shake

(c) $P(X<a)=0.9$ , then a - 90th percentile

$P(X<a)=0.9\implies P(1.184+0.587N(0,1)<a)=0.9\implies P(N(0,1)<{\frac {a-1.184} {0.587}})=0.9\implies {\frac {a-1.184} {0.587}}=1.28\implies a=1.935$

The 90th percentile is smaller than 4, which means not all of such earthquakes will cause indoor items to shake