Probability Question Normal Distribution

Question

Given that X has a normal distribution with mean 10 and standard deviation 2,find:

1. P(9 <=X <=12)
2. If P(a <= X <= b) = 0.95, then find a and b.

Answer 1

Given that X has a normal distribution with mean 10 and standard deviation 2,find:

1.  P(9 <=X <=12)

Step 1: Find the Z scores for 9 and 12.

z1 = (9-10)/2 = -1/2

z2 = (12-10)/2 = 1

Step 2: Use table or TI-84 to find area under the curve.

TI-85 normalcdf ( -1/2, 1) = .5328

Table: Area (1) - Area (-1/2) using standard z tables

Area = .8413 - ..3085 = .5328

2  If P(a <= X <= b) = 0.95, then find a and b.

Step 1: Find the z-scores for Areas of 0.025 and 0.975

z1 = -1.96 and z2 = 1.96

Step 2: Find the x-score for each

x1 = (-1.96)(2) + 10 = 6.08

x2 = (1.96)(2) + 10 = 13.92

Answer 2

Φ(z) is the CDF of Z, where Z ~ N(0, 1)

Then, as X is distributed normally with mean 10 and standard deviation 2,

P(9 <= X <= 12) = P((9-10)/2 <= Z <= (12-10)/2) = P(-.5 <= Z <= 1) = Φ(1) - Φ(-.5)

We may get values of Φ either from a table or statistical software, for example, from a statistical calculator or Excel. In Excel, we get norm.s.dist(1,1) - norm.s.dist(-.5,1) = .8413 - .3085 = .5328

For P(a<= X <= b) = .95, we get, for .95 <= y <= 1,

a = 10 + 2* Φ^-1(y - .95)

b = 10 + 2* Φ^-1(y)

A popular answer for this, for a and b to be symmetric about 10, will be, letting y = .975

a = 10 + 2* Φ^-1(.025) = in Excel, 10 + 2* norm.s.inv(.025) = 6.0801 or 6.08

b = 10 + 2* Φ^-1(.975) = in Excel, 10 + 2* norm.s.inv(.975) = 13.9199 or 13.92

Two other values, equally correct, would be, letting y = .95,

a = 10 + 2* Φ^-1(0) = -∞

b = 10 + 2* Φ^-1(.95) = 13.2897 or 13.29

Two other values, equally correct, would be, letting y = 1,

a = 10 + 2* Φ^-1(.05) = 6.7103 or 6.71

b = 10 + 2* Φ^-1(1) = ∞