Question

3.   The Kaufman Assessment battery for children is designed to measure achievement and intelligence with a special emphasis on nonverbal intelligence. Its global measures, such as its Sequential Processing score, are scaled to have a mean of 100 and a standard deviation of 15. Assume that the Sequential Processing score has a normal distribution.

b.   What is the proportion of children will have Sequential Processing score between 90 and 110?

c.   In a sample of 40 children, what is the probability the sample mean will differ from the population mean by more than 3 points (either positive or negative).

Answer 1

Solution:
Given in the question
Mean. =100
Standard deviation = 15
Solution(b)
We need to find P(90<Xbar<110) = P(X<110)-P(X<90)
Z = (90-100)/15 = -0.67
Z = (110-100)/15 = 0.67
From Z table we found p-value as follows:
P(90<Xbar<110) = P(X<110)-P(X<90) = 0.7486 - 0.2514 = 0.4972

Solution(c)
No. of sample = 40
P(Xbar<97 or Xbar>103) = P(Xbar<97) + 1 - P(Xbar<103)
Z = (103-100)/15/sqrt(40) = 3/15/sqrt(40) = 1.24
Z = (97-100)/15/sqrt(40) = -3/15/sqrt(40) = -1.24
From Z table we found
P(Xbar<97 or Xbar>103) = P(Xbar<97) + 1 - P(Xbar<103) = 0.1075 +1 - 0.8925 = 0.215