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Reminders • Email me and the TAs today if you have any questions/concerns about grading of quiz 4 or HW 4 1 Warm Up • Final grades in BIO 180 at UW are Normally distributed with a mean of 3.0 and a SD of 0.3. Final grades in STAT 566 are Normally distributed with a mean of 3.7 and a SD of 0.1. If you get a 3.4 in BIO 180 and a 3.8 in STAT 566, in which course is your standard score higher? 2 Chapter 13 part 2: Normal calculations Aaron Zimmerman STAT 220 - Summer 2014 Department of Statistics University of Washington - Seattle 3 Percentiles • The cth percentile of a distribution is the value such that c percent of the observations lie below it and the rest lie above it ? If you score at the 65th percentile on a test, then 65% of the students who took the exam scored lower than you and the other 35% scored higher ? If your height is at the 20th percentile within your gender, then 20% of all people in your gender are shorter than you and the rest are taller Key idea #1: Each percentile in a normal distribution corresponds with a standard score 4 Percentiles • Table B in the back of your book allows you to “convert” between standard scores and percentiles ? For example, a standard score of -2.0 corresponds with the 2.27th percentile (2.27% of observations in Normally distributed data have a standard score less than -2.0) ? We’ve already seen this one, but the table will allow you to do it for percents other than 68-95-99.7 Key idea #2: Percentiles can be interpreted as the area under the standard normal curve to the left of the standard score 5 How to use a Standard Score Table 6 How to use a Standard Score Table Check how the table works! You can either: • (1) Find a percentile given a score ? e.g. What percentile is associated with a score of -0.5? • (2) Find a score given a percentile ? What standard score has 30% of standard scores less than it? 7 How to use a Standard Score Table 8 ACT scores • Remember that ACT math scores are normally distributed with a mean of 18 and a SD of 6 • Grainne had a standard score of 1.5 while Karthik had a standard score of -0.5 on the ACT math test • What were their math scores? 9 ACT scores • The percentile of Karthik’s score is the area to the left of his standard score in the standard normal distribution ? If you look up -0.5 in Table B, you see that Karthik scored at the 30.85th percentile on the ACT math test. 10 ACT scores • The percentile of Grainne’s score is the area to the left of her standard score in the standard normal distribution ? If you look up 1.5 in Table B, you see that Grainne scored at the 93.32th percentile on the ACT math test. 11 What percentile corresponds to a standard score of -1.2? 12 What percentile corresponds to a standard score of -1.2? Answer: 11.51 percentile 13 What standard score corresponds to the 90th percentile? 14 What standard score corresponds to the 90th percentile? Answer: Approximately a score of 1.3 15 What percent of standard scores are between 0.5 and 1? 16 What percent of standard scores are between 0.5 and 1? Answer: 84.13 − 69.15 = 14.98 % 17 What % of scores are larger than a standard score of 0.3? 18 What % of scores are larger than a standard score of 0.3? Answer: 1 − 61.79 = 38.21 % 19 Strategy for problems involving percentiles of Normal distributions • Find the relevant mean • Find the relevant standard deviation • If you have an observation and want to calculate the percentile (the percent below that value): ? Calculate the standard score ? Use Table B to convert to a percentile ? If you need the percent above the observation, calculate the percentile and subtract from 100 20 Distribution of heights The heights of men are Normally distributed with a mean of 70 in. and a SD of 2.5 in. The heights of women are Normally distributed with a mean of 66 in. and a SD of 3 in. • If Karthik is 75 in. tall, what percentile is his height? ? Relevant mean: 70 in. ? Relevant SD: 2.5 in. ? Standard score: 75−70 =2 2.5 • Percentile: 97.73 • Conclusion: 97.73% of men are shorter than Karthik 21 Distribution of heights The heights of men are Normally distributed with a mean of 70 in. and a SD of 2.5 in. The heights of women are Normally distributed with a mean of 66 in. and a SD of 3 in. • If my little sister is 58.5 in. tall, what percent of women are taller than her? ? Relevant mean: 66 in. ? Relevant SD: 3.0 in. ? Standard score: 58.5−66 = −2.5 3.0 • Percentile: 0.62 • Conclusion: 1-0.62 = 99.38% of women are taller than Eliana 22 Heights of maple trees are Normally distributed with a mean of 34 ft. and a standard deviation of 9 ft., while the heights of elm trees are Normally distributed with a mean of 19 ft. and a standard deviation of 3 ft. What percentile is an elm tree that is 20.5 ft. tall? 23 Heights of maple trees are Normally distributed with a mean of 34 ft. and a standard deviation of 9 ft., while the heights of elm trees are Normally distributed with a mean of 19 ft. and a standard deviation of 3 ft. What percent of maple trees are taller than 25 ft.? 24 Strategy for problems involving percentiles of Normal distributions • Find the relevant mean • Find the relevant standard deviation • If you have an observation and want to calculate the percentile (the percent below that value): ? Calculate the standard score ? Use Table B to convert to a percentile ? If you need the percent above the observation, calculate the percentile and subtract from 100 • If you have a percentile and need the observation value: ? Find the standard score in Table B for that percentile ? Work backwards: observation = (standard score) × SD + mean 25 Distribution of heights The heights of men are Normally distributed with a mean of 70 in. and a SD of 2.5 in. The heights of women are Normally distributed with a mean of 66 in. and a SD of 3 in. • If you are a woman, how tall must you be to be at the 90th percentile? ? Relevant mean: 66 in. ? Relevant SD: 3.0 in. ? Percentile: 90 ? Standard score: about 1.3 • obs = (1.3) × 3 + 66 = 69.9 • Conclusion: A woman must be 69.9 in. tall to be at the 90th percentile 26 Distribution of heights The heights of men are Normally distributed with a mean of 70 in. and a SD of 2.5 in. The heights of women are Normally distributed with a mean of 66 in. and a SD of 3 in. • If you are a man, how tall must you be to be in the top 35% of men’s heights? ? Relevant mean: 70 in. ? Relevant SD: 2.5 in. ? Percentile: 65 ? Standard score: about 0.4 • obs = (0.4) × 2.5 + 70 = 71 • Conclusion: A man must be 71 in. tall to be in the top 35% of men’s heights 27 Heights of maple trees are Normally distributed with a mean of 34 ft. and a standard deviation of 9 ft., while the heights of elm trees are Normally distributed with a mean of 19 ft. and a standard deviation of 3 ft. How tall is an elm tree at the 10th percentile? 28 Heights of maple trees are Normally distributed with a mean of 34 ft. and a standard deviation of 9 ft., while the heights of elm trees are Normally distributed with a mean of 19 ft. and a standard deviation of 3 ft. How tall does a maple tree need to be to be in the top 1 % of all maple trees? 29 Homework #5 • Finish reading Chapter 13 if you haven’t already • Do problems 13.21, 13.22, 13.23, 13.24, 13.25, 13.27, 13.29, 13.30 30