Question

[1] A RESEARCHER DISCOVERED FROM A SAMPLE OF 750 STUDENTS WHO TOOK AN I.Q. TEST THAT THE SAMPLE MEAN WAS 100 AND THE SAMPLE STANDARD DEVISTION WAS 15.
[A] IF A STUDENT HAD AN I.Q. SCORE OF 110; BY WHAT PERCENTAGE IS HIS SCORE BETTER THEN ANYONE ELSE.
[B] HOW MANY STUDENTS HAD I.Q. SCORES ABOVE 140.
[C] WHAT IS THE PERCENTAGE OF STUDENTS THAT HAD I.Q. SCORES BETWEEN 90 AND 100.
[D] WHERE WILL THE LOWEST I.Q. SCORE FALL.
[E] IF THE SAMPLE SIZE WAS CHANGED TO 1,000 STUDENTS. WOULD THE ANSWER IN PART [B] ABOVE BE DIFFERENT. EXPLAIN YOUR ANSWER.

Answer 1

solution:

according to central limit theorem if sample size is large the sampling distribution is approximately normal.

so, the mean of sampling distribution =

standard deviation =

a)

if IQ score(X) = 110,

P(X < 110) = value of z to the left of 0.67 = 0.7486

so, the IQ score of 110 is better than 74.86% of IQ scores

b)

P(X > 140)

P(X > 140) = 1 - value of z to the left of 2.67 = 1 - 0.9962 = 0.0038

the number of students has IQ score greater than 140 = 750*0.0038 = 2.85 = approximately 3

c)

P(90 < X < 100)

for X = 90,

for X = 100,

P(90 < X < 100) = (value of z to the left of 0) - ( value of z to the left of -0.67) = 0.5 - 0.2514 = 0.2486

percentage of studets having IQ score between 90 and 100 = 24.86%

d)

the lowers IQ score will fall with probability approximately equal to 0 to the left side of the distribution.

e)

if the sample size is changed to 1000, there will not be effeect on probability of part b if sample mean and standard deviation of sample remain same.

because as the distribution is normal the standard error remain same for a randomly selected IQ score.