Question

1) Assume X ~ N(20, 25)

a) Find P(X > 25)

b)The value of x if P(X > x) = 0.975

c) Find the values of a and b, two symmetrical values about 20 such that P(a < X < b) = 0.975

Answer 1

X ~ N(20, 25)

a) P(X > 25) = P( (X - 20 )/ 5 > (25 - 20 )/ 5) using standardization , Z = X - mean / Standard deviation

P(Z > 1 ) = 1 - P( Z < 1 ) = 1 - 0.8413 = 0.1587 ( from normal table )

b) If , P(X > x) = 0.975

then, P(X < x) = 1 - 0.975 = 0.025

therefore from normal table x = -1.96

or, P(X > - 1.96 ) = 0.975

c)  P(a < X < b) = 0.975 and values are symmetric

P( (a - 20 )/ 5 < ( X - 20 )/ 5 < ( b- 20 )/ 5 ) = 0.975

P( (a - 20 )/ 5 < Z < ( b- 20 )/ 5 ) = 0.975 (1)

Also, we know that

P( -2.241 < Z < 2.241 ) = 0.975 ( from normal table ) (2)

equating 1 and 2

(a - 20 )/ 5 = -2.241

a - 20 = -11.205

a = 20 - 11.205 = 8.795

(b - 20 )/ 5 = 2.241

b - 20 = 11.205

b = 20 + 11.205 = 31.205