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Statistics 2-5: Z-Score (Standard Score) Practice/Review One of the most common ways to use a z-score is to compare values from different data sets. x The z-score, or standard score, is found by using the formula z The number that you get can be positive or negative; it is the number of standard deviations the original entry is from the mean. *A positive z-score means that the original value is greater than the mean. *A negative z-score means that the original value is less than the mean. *A z-score of zero means that the original value is the mean. *A z-score for a data value that is more than 2 standard deviations beyond the mean tells us that that data value is an ________________. 1. With a height of 75 inches, Lyndon Johnson was the tallest president of the past century. With a height of 85 inches, Shaquille O’Neal was the tallest player on the Boston Celtics before he retired. Who is relatively taller: Lyndon Johnson among the presidents of the past century, or Shaquille O’Neal among the players on his Boston Celtics team? Presidents of the past century have heights with a mean of 71.5 inches and a standard deviation of 2.1 inches. Basketball players for the Boston Celtics have heights with a mean of 80.0 inches and a standard deviation of 3.3 inches. Are either of these heights considered “unusual” compared to their group? 2. William McKinley was the shortest president of the last century, with a height of 67 inches. Use the mean and standard deviation from #1 to do parts (a) and (b) a. Calculate McKinley’s z-score. b. Is McKinley’s height considered unusual? Why or why not? 3. The Beanstalk club is limited to women and men who are very tall. The minimum height requirement for women is 70 inches. Women’s heights have a mean of 63.6 inches with a standard deviation of 2.5 inches. a. Find the z-score corresponding to a woman with a height of 70 inches and determine whether that height is unusual? b. What would the z-score be for a woman with a height of 68 inches? Is it unusual? 4. Which is relatively better: a score of 85 on a psychology test or a score of 48 on an economics test? Scores on the psychology test have a mean of 90 and a standard deviation of 10. Scores on the economics test have a mean of 55 and a standard deviation of 5. 5. You take three tests in one week. Which one did you perform better on? A) A score of 144 on a test with a mean of 128 and a standard deviation of 34. B) A score of 90 on a test with a mean of 86 and a standard deviation of 18. C) A score of 18 on a test with a mean of 15 and a standard deviation of 5. 6. Interpret this statement: A child scored in the 78th percentile on an IQ test. Use the data set below to answer questions 7 – 9. Number of absences for 10 students in my first hour: 3, 0, 7, 19, 1, 4, 5, 0, 4, 2 7. Find the 5-number summary of the data 8. Draw a box and whisker plot of the data. 9. Using the IQR definition of outlier, determine if any of the data values are outliers. Use the box and whisker plot about test score data to answer questions 10 – 13. 10. List the values for min, max, Q1 , median, and Q3 (to your best estimate). 11. What percent of data is above Q1 ? 12. What percent of data is below the median? 13. How would you describe the shape of this box and whisker plot?