. Now the following problems are quantile problems which ask for values of x on

Question

. Now the following problems are quantile problems which ask for values of x on the x-axis for which certain probabilities are true. Make sure you draw bell curves and shade in regions just like the procedure I showed you in class to solve the following problems. Suppose that Z N(0, 1) and X N(µ = 2, = 3).

(a) Find the value of a such that P(Z < a) = 0.648

(b) Find the value of b such that P(X < b) = 0.3

(c) Find the 20th percentile of X or the value b such that P(X < b) = 0.2

(d) Find the 20th percentile of Z and then use X = Z + µ (shifting and stretching) to find the 20th percentile of X. Did you get the same answer?

(e) Find the value c such that P(Z > c) 0.1.

(f) Find the value of b such that P(|Z| < b) = 0.95

(g) Find the value of b such that P(|Z| < b) = 0.95

(h) Find the value c such that P((X < 2) (X > c)) = 0.95

Explanation / Answer

mean = 2 , s = 3

In order to solve below problems, we need to find value of z and then apply central limit theorem.

a) P(Z < a) = 0.648
Here, area is given, we need to find value of a

Corresponding value of z = 0.379 (use standard z table in order to find this value)

As per central limit theorem, z = (x - mu)/s

x = mu + z*s = 2 + 0.379*3 = 3.137

we get a = 3.137

b) P(X < b) = 0.3
Here, area is given, we need to find value of b

Here z = -0.524

b = 2 -0.524*3
we get b = 0.428

c) P(X < b) = 0.2
Here, area is given, we need to find value of b

Here z = -0.841

b = 2 -0.841*3

we get b = -0.524

e)
P(Z > c) = 0.1
Here, area is given, we need to find value of c

Here z = 1.282

b = 2 + 1.282*3

we get c = 5.846

f)
P(|Z| < b) = 0.95
Here, area is given, we need to find value of b

Here z = 1.645

b = 2 + 1.645*3

we get b = 6.936

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