A researcher develops a test for selecting intellectually gi

A researcher develops a test for selecting intellectually gifted children, with a of 56 and a X of 8. (a) What percentage of children are expected to score below 60? (b) What percentage of scores will be above 54? (c) A gifted child is defined as being in the top 20%. What is the minimum test score needed to qualify as gifted?)

Using the test in question 23, you measure 64 children, obtaining a X of 57.28. Slug says that because this X is so close to the of 56, this sample could hardly be considered gifted. (a) Perform the appropriate statistical procedure to determine whether he is correct. (b) In what percentage of the top scores is this sample mean?

Solution

Normal Distribution
Mean ( u ) =56
Standard Deviation ( sd )=8
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 60) = (60-56)/8
= 4/8= 0.5
= P ( Z <0.5) From Standard Normal Table
= 0.6915 ~ 69.15%

b)
P(X > 54) = (54-56)/8
= -2/8 = -0.25
= P ( Z >-0.25) From Standard Normal Table
= 0.5987 ~ 59.87%

c)
P ( Z > x ) = 0.2
Value of z to the cumulative probability of 0.2 from normal table is 0.84
P( x-u/ (s.d) > x - 56/8) = 0.2
That is, ( x - 56/8) = 0.84
--> x = 0.84 * 8+56 = 62.736