Let the random variable X follow a normal distribution with = 50 and 2 = 64:

Find a symmetric interval about the mean for which the probability of X being in this interval is 0.95.

SOLUTION

This exercise is solved similarly to the previous one. Since the requested interval should be symmetric, half of 0.475 is on each side of the mean. For the bottom 2.5% (obtain by 0.5-0.475), the equation is P(x x1) = 0.025. By using the standardized variable we obtain: (1) P(z < x150 64 ) = 0.025 The critical value corresponding to the area of 0.025 is -1.96. We have: (2) P(z < 1.96) = 0.025 From (1) and (2) we obtain: x150 64 = 1.96 x1 = 50 + (1.96)( 64) = 34.32 For the top 97.5%, the equation is P(x > x2) = 0.975. Now (3) P(z > x250 64 ) = 0.975 The critical value corresponding to the area of 0.975 is 1.96 (note, it is symmetrical to the value corresponding to the bottom 2.5% so it just has the positive sign but the same value). Hence, we have: (4) P(x > 1.96) = 0.975 From (3) and (4) we obtain: x250 64 = 1.96 x2 = 50 + (1.96)( 64) = 65.68 Therefore, the two values are 34.32 and 65.68

I DONT UNDESRTAND THE SOLUTION COULD YOU EXPLAIN ME PLEASE, STEP BY STEP