Q:

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

Accepted Solution

A:
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 [tex]\mu=26[/tex], Standard deviation [tex]\sigma=2[/tex]Using Z-score formula,[tex]Z=\frac{\bar{x}-\mu}{\sigma}[/tex]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 [tex]\bar{x}=23[/tex] is given by,[tex]Z=\frac{23-26}{2}[/tex][tex]Z=\frac{-3}{2}[/tex][tex]Z=-1.5[/tex]The probability of getting a sentence below that is [tex]P(X<23)[/tex]i.e. [tex]P(X<23)=P(Z<-1.5)[/tex] [tex]P(X<23)=0.067[/tex]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 [tex]\bar{x}=30[/tex] is given by,[tex]Z=\frac{30-26}{2}[/tex][tex]Z=\frac{4}{2}[/tex][tex]Z=2[/tex]  The probability of getting a sentence below that is [tex]P(X\geq 30)[/tex]i.e. [tex]P(X\geq 30)=1-P(Z\leq 2)[/tex][tex]P(X\geq 30)=1-0.977[/tex][tex]P(X\geq 30)=0.023[/tex]Into percentage, [tex]P(X\geq 30)=2.3\%[/tex]3) Would you consider a 30 month sentence harsh? ExplainSince the percentage for 30 months or greater is harsh because the smaller value of percentage of [tex]P(X\geq 30)=2.3\%[/tex]