Question

Assume that a population is normally distributed with a population mean of μ = 8 and a population standard deviation of s = 2. [Notation: X ~ N(8,2) ] Use the Unit Normal Table to answer the following questions. (I suggest you draw a picture of the normal curve when answering these questions.)

8. Let X=4 Compute the z-score of X. For that z-score:

a. What proportion of the area under the Unit Normal Curve is in the tail? b. What proportion of the area under the Unit Normal Curve is in the body? c. What proportion of the area under the Unit Normal Curve is between the

mean and z?

d. What proportion of scores in the population is in the tail?

e. What proportion of scores in the population is in the body?

f. What proportion of scores in the population is between the mean and X=4? g. What proportion of scores in the population is greater than X=4?

h. What proportion of scores in the population is less than X=4?

i. Answer (d) through (h) again, but report percentages instead of proportions. j. If you randomly sample 100 scores from the population, what is the best

estimate for:

i. the number of scores in the sample that are greater than X=4?

ii. the number of scores in the sample that are less than X=4?

(For (i) and (ii), you can round your answer to the nearest whole number)

Answer 1

The z-score for X= 4 is

(a)

Area left to z-score is:

P(z < -2) = 0.0228

(b)

The proportion of the area under the Unit Normal Curve is in the body is

1 - P(z <-2) = 1 - 0.0228 = 0.9772

c)

The proportion of the area under the Unit Normal Curve is between the mean and z is

P(-2 < z < 0) = P(z <0) - P(z-2) = 0.5 -0.0228 = 0.4772

d)

The  proportion of scores in the population is in the tail is

P(X <4) = P(z < -2) = 0.0228

e)

The proportion of scores in the population is in the body is

P(X>4) = 1-P(X<4) = 1 - P(z <-2) = 1 - 0.0228 = 0.9772

f)

The proportion of the area under the Unit Normal Curve is between the mean and z is

P(4<X<8) = P(-2 < z < 0) = P(z <0) - P(z-2) = 0.5 -0.0228 = 0.4772

g)

The  proportion of scores in the population is greater than X=4 is

P(X>4) = 1-P(X<4) = 1 - P(z <-2) = 1 - 0.0228 = 0.9772

h)

The proportion of scores in the population is less than X=4 is

P(X <4) = P(z < -2) = 0.0228

(i)

From part d: 2.28%

From part e: 97.72%

From part f: 47.72%

From part g: 97.72%

From part h: 2.28%

j)

(i)

Using part g:

100 * 0.9772 = 97.72

Answer: 98

(ii)

Using part h:

100 * 0.0228 = 2.28

Answer: 3