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=7 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=7? g. What proportion of scores in the population is greater than X=7?

h. What proportion of scores in the population is less than X=7?

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=7?

ii. the number of scores in the sample that are less than X=7?

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

Answer 1

Given that it is normal fdistribution and Mean =8 and Standard deviation s=2 hence

According to question

8. At X=7, Z score Calculated as

a. To find the proportion of area in the tail below Z =-0.5 we look into the Z table as

Hence area Below Z=-0.5 is the tail area computed as 0.3085 hence proportion = 0.3085

b.As the total area of unit curve is 1 hence the are in the table can be calculated as

1-0.3085= 0.6915

c.As the area at the mean, as below the mean is 0.50 and above the mean is 0.50

Hence the area between at Z=-0.5 and Z=0 as

0.50-0.3085=0.1915

d. What proportion of Scores in the population is in the tail at below Z=-0.5

Since the percentile of scores at Z=-0.5 is 30.85th hence 0.3085 proportion of score lie in the tail.

e. Since the percentile at Z=-0.5 is 30.85 hence the score which have more are 1-0.3085=0.6915 hence 0.6915 propotion of data lies in body.

f. Proportion of score lies between X=7 and the mean as 0.1915.

g. The proportion of score at above X=7 means the proportion in the body hence 0.6915 is the proportion.

h.Since the proportion of Score above X=7 is 0.6915 hence scores below is 1-0.6915=0.3085