Suppose that X ∼ N(10,16) with mean 10 and variance 16. Please show all steps to find

Question

(a) P(9.8≤X<10.2).

(b) a so that P(X>a)≈0.5478.

(c) a so that P(X<a)≈0.1515.

Answer 1

Let us start by understanding the distribution. We are given that μ=10 and σ²=16. From this, we may deduce that μ=10 and σ=4.

Let us now look at (a). We shall start by sketching (not an exact drawing) of what the question is asking to make sure we are precisely solving for what it asks.

Now that we may see what we are solving for, let's find the z-scores.

z = (x-μ)/σ = (9.8-10)/4 = -.05

z = (x-μ)/σ = (10.2-10)/4 = .05

p (9.8 < X < 10.2) ⇒ p (-.05 < Z < .05) = p (Z<.05) - p (Z < -0.05) =.0399

Let us now look at (b). We shall start by sketching (not an exact drawing) of what the question is asking to make sure we are precisely solving for what it asks.

So for this, we need to go from probability to z-score.

Most z-tables go from negative infinity to the z-score, so we must find the area that is not shaded in this case and use that.

P(x<a) = 1-P(x>a) = 1-.5478 = .4522

This gives a z=score of -0.1201

z = (a-μ)/σ ⇒ a = zσ+μ = -0.1201*4+10 = 9.5196

Let us now look at (c). We shall start by sketching (not an exact drawing) of what the question is asking to make sure we are precisely solving for what it asks.

This gives a z=score of -1.0300

z = (a-μ)/σ ⇒ a = zσ+μ = -1.0300*4+10 = 5.8800