Question

1. Practicing z-scores.
   1. You have a variable distributed N(2,8). What is the range of values within which 68% of my observations fall?
   2. You have a variable distributed N(2,8). What is the z-score of the value 2?
   3. You have a variable distributed N(2,8). What is the z-score of the value 8?
   4. What percent of the data are to the left of 2?
   5. What percent of the data are to the left of 8?
   6. What percent of the data are between 2 and 8?

Answer 1

Solution

Back-up Theory

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ², then, Z = (X - µ)/σ ~ N(0, 1), i.e., Standard Normal Distribution ………………………..(1)

Equivalently, Z-score corresponding to X is: (X - µ)/σ ………………………….. (1a)

and hence X corresponding to Z-score is: X = µ + Zσ ………………………….. (1b)

P(X ≤ or ≥ t) = P[{(X - µ)/σ} ≤ or ≥ {(t - µ)/σ}] = P[Z ≤ or ≥ {(t - µ)/σ}] .………(2)

Probability values for the Standard Normal Variable, Z, can be directly read off from

Standard Normal Tables ……………………………………………………………… (3a)

or can be found using Excel Function: Statistical, NORMSDIST(z) which gives P(Z ≤ z) ………(3b)

Percentage points of N(0, 1) can be found using Excel Function: Statistical, NORMSINV(Probability) which gives values of t for which P(Z ≤ t) = given probability………………. ………(3c)

Since Z distribution, i.e., N(0, 1) is symmetric,

P(Z < - t) = P(Z >T) …………………………………………………………….……. (4a)

P(- t < Z < t) = 2{1 - P(Z > t)} = 2{1 - P(Z > - t)}   …………………………….……. (4a)

P(Z < 0) = P(Z > 0) == ½ ……………………………………………………………….(4c)

Empirical rule, also known as 68 – 95 – 99.7 percent rule:

P{( µ - σ) ≤ X ≤ ( µ + σ)} = 0.68; ……………………………………………….(7a)

P{( µ - 2σ) ≤ X ≤ ( µ + 2σ)} = 0.95; ……………………………………………….(7b)

P{( µ - 3σ) ≤ X ≤ ( µ + 3σ)} = 0.997 ……………………………………………….(7c)

Now to work out the solution,

Let X ~ N(2, 8); …………………………………………………………………………… (8)

NOTE: 8 is taken as the standard deviation and not variance.

Part (a)

Vide (7a) and (8), the range of values within which 68% of my observations fall

= 2 – 8 and 2 + 8

= [- 6, 10]    Answer 1

Part (b)

Vide (1a) and (8), z-score of the value 2

= (2 – 2)/2.8284

= 0 Answer 2

Part (c)

Vide (1a) and (8), z-score of the value 8

= (8 – 2)/8

= 0.75 Answer 3

Part (d)

Since mean is 2, vide (4c) and (8), probability to the left of 2 is 0.5. Hence,

percent of the data are to the left of 2 = 100 x ½ = 50% Answer 4

Part (e)

Percent of the data are to the left of 8

= 100 x P(X < 8)

= 100 x P{Z < (8 – 2)/8}

= 100 x P(Z < 0.75)

= 100 x 0.7614 [vide (3a)]

= 76.14% Answer 5

Part (f)

Percent of the data are between 2 and 8

= Percent of the data are to the left of 8 - Percent of the data are to the left of 2

= 76.14 – 50 [vide Answers 4 and 5]

= 26.14   Answer 6

DONE