Question

The prosecutor's examination of court records in a specific county shows that the mean sentence length for first-offense drug dealers is 23 months with a standard deviation of 2.5 months. The records show that the sentence lengths are normally distributed. (You will need to use the Z table  or software for this question.)

1) What percentage of first-time convictions for drug dealing are 20 or fewer months in length?  %

b) A defense attorney is concerned that his client's sentence was unusually harsh at 28 months. What percent of sentences are 28 months or longer?   %

c) What is the sentence length that separates top (longest) 2.5% of the lengths from the rest?  months

Answer 1

Solution :

Given that ,

a) P(x 20 )

= P[(x - ) / (20 - 23) / 2.5]

= P(z -1.20)

Using z table,

= 0.1151

percentage = 11.51%

b) P(x 28 ) = 1 - P(x   28)

= 1 - P[(x - ) / (28 - 23) / 2.5 ]

= 1 -  P(z 2.0)

= 1 - 0.9772

= 0.0228

percent = 2.28%

c) Using standard normal table,

P(Z > z) = 2.5%

= 1 - P(Z < z) = 0.025  

= P(Z < z) = 1 - 0.025

= P(Z < z ) = 0.975

= P(Z < 1.96 ) = 0.975  

z = 1.96

Using z-score formula,

x = z * +

x = 1.96 * 2.5 + 23

x = 27.9 months

x = 28 months