Question

Examination of court records in a particular state shows that the mean sentence length for first-offense drug dealers is 26 months with a standard deviation of 2 months. The records show that the sentence lengths are normally distributed. 1) What is the Z score for a 23 month sentence length? What is the probability of getting a sentence below that? 2) A defense attorney is concerned that his client's sentence was unusually harsh at 30 months. What percent of sentences are 30 months or longer? Calculate the Z score and report the area. 3) Would you consider a 30 month sentence harsh? Explain

Answer 1

Answer and explanation:

Given : Examination of court records in a particular state shows that the mean sentence length for first-offense drug dealers is 26 months with a standard deviation of 2 months. The records show that the sentence lengths are normally distributed.

i.e. Mean
`\mu=26`, Standard deviation
`\sigma=2`

Using Z-score formula,

`Z=\frac{\bar{x}-\mu}{\sigma}`

1) What is the Z score for a 23 month sentence length? What is the probability of getting a sentence below that?

The Z score value for a
`\bar{x}=23` is given by,

`Z=(23-26)/(2)`

`Z=(-3)/(2)`

`Z=-1.5`

The probability of getting a sentence below that is
`P(X<23)`

i.e.
`P(X<23)=P(Z<-1.5)`

`P(X<23)=0.067`

2) A defense attorney is concerned that his client's sentence was unusually harsh at 30 months. What percent of sentences are 30 months or longer? Calculate the Z score and report the area.

The Z score value for a
`\bar{x}=30` is given by,

`Z=(30-26)/(2)`

`Z=(4)/(2)`

`Z=2`

The probability of getting a sentence below that is
`P(X\geq 30)`

i.e.
`P(X\geq 30)=1-P(Z\leq 2)`

`P(X\geq 30)=1-0.977`

`P(X\geq 30)=0.023`

Into percentage,
`P(X\geq 30)=2.3\%`

3) Would you consider a 30 month sentence harsh? Explain

Since the percentage for 30 months or greater is harsh because the smaller value of percentage of
`P(X\geq 30)=2.3\%`