Computing z Scores

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.

  1. 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.
  2. Calculate the z-score for Police officer #1. Was this score high enough to be considered for promotion?
  3. Calculate the z-score for Police officer #2. Was this score high enough to be considered for promotion?
  4. 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.
  5. What percentage of first time convictions for drug dealing receive 24 or fewer months?
  6. 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?
  7. 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?
  8. 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.
  9. Calculate the mean and standard deviation for the number of incidents of shoplifting.
  10. What is the z-score for an addict that has shoplifted 6 times?
  11. 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?
  12. Find the z-score for an addict who shoplifted once (1) in the past week.
  13. What percentage of addicts shoplifted 1 or fewer times in the past week?
  14. 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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