1) (75 points) Normal Distribution and Central Limit Theorem In the year 2028, A

Question

1) (75 points) Normal Distribution and Central Limit Theorem In the year 2028, Araceli Gomez is a leading expert in the field of criminal justice psychology. Araceli is researching the relationship between sub violence. Suppose the length of time that individuals have been substance abusers distributed with a mean of 12.7 years and a standard deviation of 3.2 years. and stance abuse and domestic is normally la) What is 60th percentile for the length of time of substance abuse? lb) What is the probability a randomly selected person will have been a substance abuser for more than 10 years? 1c) In a random sample of 4 substance abusers, what is the probability the sample mean is more than 10 years? 1d) As Araceli discusses the joy of solving statistical problems, a colleague deseribes a statistical question regarding an individual on trial for domestic violence. The time the individual was a substance abuser was 1.32 standard deviations below the mean. How long was this person a substance abuser?

Explanation / Answer

Solution1a:

60 percentile to z

z=0.253

z=x-mean/sd

0.253=x-12.7/3.2

x=0.253*3.2+12.7

x=13.5096

60 th percentile=13.5096

Solution1b:

P(X>10)

z=x-mean/sd

=10-12.7/3.2

z=-0.844

P(Z>-0.844

=1-P(Z<0.844)

=0.7995

ANSWER:0.7995

Solutionc:

P(X bar>10)

z=x-mean/sd/sqrt(n)

n=4

z=10-12.7/3.2/sqrt(4)

z=-1.6875

P(z>-1.688)

=P(z<1.688)

=0.9545

ANSWER:0.9545

Solution1d:

z=x-mean/sd

z=-1.32

x-12.7/3.2=-1.32

x=-1.32*3.2+12.7

x=8.476

ANSWER:8.476

He was 8.476 years

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