For all problems: Show your steps in reasoning/computation. Draw a distribution,

Question

For all problems: Show your steps in reasoning/computation. Draw a distribution, mark the t. score and/or area of question when appropriate (no need to turn it in). Round the results to two decimal points. I. Prison officials collect data for length of inmate prison term, in months, for various crimes. The data, which are approximately normally distributed, are listed below The following convictions and sentences have just been issued: John, robbery. 62 months Pete, manslaughter, 123 months Clyde, assault. 43 months A ¢ Maurice, drug dealing, 47 months Relatively, which convict received the harshest sentence? 2. John scores 124 on an IQ test. He scores 6.2 on a reading achievement test. He scores 71 on math achievement lest. The means and standard deviations for the three tests are as follows: What percentage of the population would be expected to score LOWER than John in IQ? What percentage of the population would be expected to score between the mean and John's IQ score? Given John's IQ. is his READING performance better, worse, or the same as you would expect? Given John's IQ. is his MATH performance better, worse, or the same as you would expect? Transform John's reading and math scores to the same scale as IQ. that is. M = 100. SD = 16.

Explanation / Answer

Question 1

Solution:

First of all we have to find the z-scores for the given convicts. The Z-score formula is given as below:

Z = (X – mean) / SD

The z-score for John is given as below:

Z = (62 – 58) / 7 = 0.571428571

The z-score for Pete is given as below:

Z = (123 – 112) / 13 = 0.846153846

The z-score for Clyde is given as below:

Z = (43 – 38) / 4 = 1.25

The z-score for Maurice is given as below:

Z = (47 – 43) / 5 = 0.8

From the above calculations, it is observed that the Z-score is highest for Clyde. So, relatively Clyde received the harshest sentence.

Question 2

Part a

We have to find P(X<124)

Z = (X – mean) / SD

Z = (124 – 100) / 16 = 1.5

P(Z<1.5) = 0.933192799

Required probability = P(x<124) = 0.933192799

Part b

Here, we have to find P(100<X<124)

P(100<X<124) = P(X<124) – P(X<100)

Z-score for x = 124

Z = (X – mean) / SD

Z = (124 – 100) / 16 = 1.5

P(Z<1.5) = 0.933192799

Z score for x = 100

Z = (100 – 100) / 16 = 0

P(Z<0) = 0.5

P(X<100) = 0.5

P(100<X<124) = P(X<124) – P(X<100)

P(100<X<124) = 0.933192799 – 0.50

P(100<X<124) = 0.433192799

Required probability = 0.433192799

Part c

Here, first of all we have to find the Z score for the John’s reading performance.

We are given mean = 6.5 and SD = 0.6

John’s reading performance = X = 6.2

Z = (X – mean) / SD

Z = (6.2 – 6.5) / 0.6

Z = -0.5

Now, we have to find Z score for John’s IQ score.

John’s IQ score = X = 124

We are given mean = 100 and SD = 16

Z = (X – mean) / SD

Z = (124 – 100) / 16

Z = 1.5

Z score for reading performance is less than Z score for IQ. So, we would expect worse reading performance of John.

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