1. Advanced Math Topics 3.8 Z-Scores

2. Two brothers are in different math classes. Their most recent test scores are shown in orange while their class is shown in yellow. 12, 15, 17, 17, 22, 24, 27, 31, 32, 32, 34, 39, 39, 40, 40 65, 72, 73, 73, 74, 74, 75, 75, 76, 76, 77, 78, 78, 79, 80 What are their percentile ranks? The first brother is in the 80 th percentile. The second brother is in the 80 th percentile. Which brother is the stronger student relative to his class? Although they have the same percentile rank, the first brother scored a lot higher than many of his fellow classmates. The second brother scored slightly higher than most of his classmates. Percentile rank shows the percentage of students the brothers scored above, but a z-score will reflect the discrepancy from the brother’s scores to everyone else’s score.

3. z-score of x = X – X s What would your z-score be if your grade was equal to the mean? Zero What type of # would your z-score be if your grade was less than the mean? Negative What type of # would your z-score be if your grade was more than the mean? Positive Thus, the higher your z-score, the better you scored relative to the other scores!

4. New York rated its top 5 pizza companies and San Francisco did the same, but they used different scales. A high score shows that people enjoyed the pizza. The results are: BrandNY Rating David’s 1 Mama Mia’s 10 Papa John’s15 Round Table 21 Pizza Time28 BrandSF Rating Domino’s 25 Pizza Hut35 Zachary’s45 Mt. Mike’s50 Pavlo’s70 Which brand has the least taste appeal? At first glance, David’s seems to be the lowest. But since there were different scales and other relative scores to consider, we must find the z-score. z-score of x = X – X s x = 1545 s = 10.3216.96 David’s z-score = 1 – 15 10.32 = -1.36 Domino’s z-score = 25 – 45 16.96 = -1.18 The worst rated pizzeria was David’s because it has the lowest z-score. It stands for how many standard deviations The score is from the mean.

5. BrandNY Rating David’s 1 Mama Mia’s 10 Papa John’s15 Round Table 21 Pizza Time28 BrandSF Rating Domino’s 25 Pizza Hut35 Zachary’s45 Mt. Mike’s50 Pavlo’s70 x = 1545 s = 10.3216.96 Which brand has the greatest taste appeal? Pizza Time z-score = 28 – 15 10.32 = 1.26 Pavlo’s z-score = 70 – 45 16.96 = 1.47 The best rated pizzeria was Pavlo’s because it has the highest z-score. 3 Rows Closest to the window: Find the z-score for Pizza Time. 3 Rows Closest to the door: Find the z-score for Pavlo’s.

6. BrandNY Rating David’s 1 Mama Mia’s 10 Papa John’s15 Round Table 21 Pizza Time28 BrandSF Rating Domino’s 25 Pizza Hut35 Zachary’s45 Mt. Mike’s50 Pavlo’s70 Z-score -1.36 -0.48 0 0.58 1.26 Z-score -1.18 -0.59 0 0.29 1.47 What do you notice? Sum = 0Sum = -0.01 It is not 0 because of rounding. In any distribution, the sum of the z-scores is 0.

7. Together: Look at Example #1 on P. 153 and Example #3 on P. 155 HW: P. 156 #1-5, 7 #3 Answer: a) Lake D, 1.41 b) Lake S, -1.54 The back of book has different #’s because they computed with a population formula rather than a sample formula, which gives slightly different data