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,

$Z=\frac{23-26}{2}$

$Z=\frac{-3}{2}$

$Z=-1.5$

The probability of getting a sentence below that is $P(X$

i.e. $P(X$

$P(X$

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,

$Z=\frac{30-26}{2}$

$Z=\frac{4}{2}$

$Z=2$  

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 $P(X\geq 30)=2.3\%$