The scores for the intelligence quotient (IQ) test are normally distributed with

Question

The scores for the intelligence quotient (IQ) test are normally distributed with the mean of 100 and standard deviation of 15.

a)What is the probability that at least two of five randomly selected people have IQ test score higher than 110?

b)A person is randomly selected from all people whose IQ are higher than 110. What is the probability that this person has IQ higher than 120?

c)One thousand people are randomly selected. Approximately what is the probability that fewer than 90 of them have IQ score higher than 120?Note: Use Normal approximation of binomial distribution

d)Determine the minimum IQ scores for the top 5 % of the population.e)Find the probability that IQ score is between 88 and 112.

f)Find the probability that IQ score is between 108 and 120.

g)If we knew our subject’s score is between 108 and 120, what will be the probability that this score fall between 88 and 112

Explanation / Answer

a. Find the z score of 110

Z=110-100/15=0.67

P(z>0.67)=0.5-P(0<z<0.67)=0.5-0.7236=0.2514

Now find P(2<=x<=5) for n=5 and p=0.2514

So P(2<=x<=5)= 0.3701

b. Z(120)=1.33

P(x>120)=0.0918

P(x>120/x>110)=0.0918/0.2514=0.3652

c. P(x>120)=0.0918

We need find P(x<90) for n=1000 and p=0.0918

P(x)=ncxp^x(1-p)^n-x

P(x<90)= 0.4060

d. P(X>x)=0.05

Using z table we get P(Z>1.645)=0.05

Hence z=x-mean/sd=1.645

So x=1.645*15+100=124.675

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