M3A1: Computing z Scores
Instructions: Complete the following questions. Enter answers and explanations in this document. Show calculations in an Excel file, using a separate worksheet (tab) for each question.
- Police departments use exams as part of their promotion process where officers have to score higher than 80% of those taking the exam to be considered for promotion. (This is a z-score of 0.83 or above). Two officers write their exams. Police officer #1 receives a grade of 70 and police officer #2 receives a grade of 72. Note that the mean (of all officers taking the exam is) is 62 and the standard deviation is 11.
- Calculate the z-score for Police officer #1. Was this score high enough to be considered for promotion?
- Calculate the z-score for Police officer #2. Was this score high enough to be considered for promotion?
- Examination of court records in a particular state shows that mean prison sentence length for first-offense drug dealers is 26 months with a standard deviation of two months. The records reveal that sentence lengths are approximately normally distributed.
- What percentage of first time convictions for drug dealing receive 24 or fewer months?
- A defense attorney was concerned that her client’s sentence was particularly harsh at 29 months. What is the probability of receiving a sentence length of 30 months or more? Would you agree with the defense attorney that 29 months is harsh? Why or why not?
- Another defense attorney is pleased that her client received a sentence of 20 months. How likely is it that a person would receive a sentence of 20 or fewer months? Do you think the defense attorney is right to be pleased? Why or why not?
- A researcher talks with 20 heroin addicts to find out how many times they have shoplifted in the past week. The data are as follows: 5, 1, 2, 0, 6, 0, 3, 8, 6, 9, 3, 0, 9, 1, 2.
- Calculate the mean and standard deviation for the number of incidents of shoplifting.
- What is the z-score for an addict that has shoplifted 6 times?
- What is the percentage of heroin addicts who have shoplifted 6 or more times in the last week? Would you say it is unusual to find heroin addicts who have shoplifted 6 or more times?
- Find the z-score for an addict who shoplifted once (1) in the past week.
- What percentage of addicts shoplifted 1 or fewer times in the past week?
- Domestic violence in the United States is a serious problem. Once arrested, offenders are likely to re-offend. Domestic violence counseling programs define success when an offender completes a counseling program and goes six or more months (180 days) without re-offending.
One study found that for men who do re-offend, the 95% confidence interval on days to re-offend is 20.79 to 46.21. Please interpret and explain in your own words what this confidence interval means.
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