Question

Let the random variable X follow a normal distribution with μ =40 and σ^2 =81.

The probability is 0.03 that X is in the symmetric interval about the mean between which two numbers?

Round to one decimal place as needed. Use ascending​ order

Answer 1

Solution :

Given that,

mean = $\mu$ = 40

Variance = $\sigma$² = 81

standard deviation = $\sigma$ = 9

P( x < X < x ) = 0.03
Middle 0.03 probability is represented by 0.03/2 = 0.0150 area each on the left side and
right side of mean because X is in the symmetric interval about the mean.
The z value which represents 0.0150 area on the left side of mean is - 0.0375 whereas that of the z value on the right side is + 0.0375
Using z score formula

z = (X - $\mu$) / $\sigma$

- 0.0375 = (X - 40)/9
- 0.3375 = X - 40
X = 39.6625
And
+ 0.0375 = (X - 40)/9
0.3375 = X - 40
X = 40.3375
The numbers required are 39.6625 and 40.3375