Normal Distribution?

Question

The random variable G has a normal distribution. mean 20, and a standard deviation o(sigma). We know that P(G > 15)=0.6. Find the value of sigma.

Answer 1

Mean of normal distribution is μ = 20 .

G is a random variable of normal distribution .

Given P(G > 15) = 0.6 .

P(G > 15) = P ( X−μ>15−20 )= P [ ( X−μ )/σ> (15−20) /σ ]

Since  Z = ( X−μ )/σ

 P [Z > (15−20) /σ ]

 P [Z > -5 /σ ]

So  P [ ( X−μ )/σ> (15−20) /σ ] = P ( Z > -5/σ )

Given P(G>15) =0.6 

P(Z > -5/σ) =0.6 

Because of the symmetry of the Z graph 

P(Z>a) = P(Z<-a) where a=5/σ 

P(Z>-5/σ) = P(Z<5/σ)=0.6

From Z Score Table 

P(Z< 0.26)=0.600 

So 5/σ = 0.26 .

σ  = 5 / 0.26

σ  =  19.2307

So the standard deviation is  σ  =  19.2307 .