Scores on the PAT (Programmer Apptitute Test) are normally distributed with a me

Question

Scores on the PAT (Programmer Apptitute Test) are normally distributed with a mean of 100 and a standard deviation of 10. Answer the following questions. Use proper mathematical notation for your answers and provide a one sentence explanation of your answer.

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If 20 people are randomly selected, find the probability that they have a mean PAT score less than 90. Include a one-sentence explanation of your answer. Notation Answer std err If 50 people are randomly selected, find the probability that they have a mean PAT score greater than 100. Include a one-sentence explanation of your answer. Notation Answer std err If 36 people are randomly selected, find the probability that their mean PAT score is between 75 and 95. Include a one sentence explanation of your answer. Answer Notation std err

Explanation / Answer

Here mean=100 and sd=10

a. We need to find P(xbar<90), for n=20

As population is normal as per central limit theorem sample mean is normal with mean=100 and sd=10/sqrt(20)=2.24

So P(xbar<90)=P(z<90-100/2.24)=P(z<-4.47)=0.5-P(-4.47<z<0)=0.5-0.5=0

2. Here n=50, and we need to find P(xbar>100) as we knowz=xbar-mean/sd/sqrt(n)

So here P(z>0)=0.5-P(0<=z<=0)=0.5-0=0.5

3. Now we need to find P(75xbar<95)=P(75-100/(10/6)<z<95-100/(10/6))=P(-15<z<-3)=0.0013

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