Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor. 3-3

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2 3-2

3 EXAMPLE Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor

4 (a) Compute the population mean and standard deviation. (b) Draw a histogram to verify the data is bellshaped. (c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (d) Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 34 and

5 (a) Using a TI-nspire graphing calculator, we find (b) 57.4 and Pearson Prentice Hall. All rights reserved 3-5

6 (c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (d) Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 34 and (look back at original data!)

7 One measure of intelligence is the Stanford-Binet Intelligence Quotient (IQ). IQ scores have bell-shaped distribution with a mean of 100 and a standard deviation of 15 A. What percentage of people has an IQ score between 70 and 130? B. What percentage of people has an IQ score less than 70 or greater than 130? C. What percentage of people has an IQ score below 85?

8 3.4 Measures of Position and Outliers

9 The Z-Score

10 EXAMPLE Using Z-Scores The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data based on information obtained from National Health and Examination Survey. Who is relatively taller? Kevin Garnett whose height is 83 inches Candace Parker whose height is 76 inches or 3-10

11 Sample Problem Score on ACT was 26 with a mean of 22 and sd of 3. Score on SAT was 950 with mean of 925 and sd of 25. Which score is "better"?

12 Quartiles divide data sets into fourths, or four equal parts. The 1 st quartile, denoted Q 1, divides the bottom 25% the data from the top 75%. Therefore, the 1 st quartile is equivalent to the 25 th percentile. The 2 nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2 nd quartile is equivalent to the 50 th percentile, which is equivalent to the median. The 3 rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3 rd quartile is equivalent to the 75 th percentile. 3-12

13 3-13

14 EXAMPLE Finding and Interpreting Quartiles A group of Brigham Young University Idaho students (Matthew Herring, Nathan Spencer, Mark Walker, and Mark Steiner) collected data on the speed of vehicles traveling through a construction zone on a state highway, where the posted speed was 25 mph. The recorded speed of 14 randomly selected vehicles is given below: 20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40 Find and interpret the quartiles for speed in the construction zone. In addition find the mean, median, and standard deviation. (using technology) 3-14

15 Interpretation: 25% of the speeds are less than or equal to the first quartile, 28 miles per hour, and 75% of the speeds are greater than 28 miles per hour. 50% of the speeds are less than or equal to the second quartile, 32.5 miles per hour, and 50% of the speeds are greater than 32.5 miles per hour. 75% of the speeds are less than or equal to the third quartile, 38 miles per hour, and 25% of the speeds are greater than 38 miles per hour. 3-15

16 Interquartile Range 3-16

17 EXAMPLE Determining and Interpreting the Interquartile Range Determine and interpret the interquartile range of the speed data. Q 1 = 28 Q 3 = 38 IQR Q Q The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour. 3-17

18 Suppose a 15 th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range? Mean Median Standard Deviation IQR With Out 15 th Car With 15 th Car Which measures should we report now? When we add the 15th car which changes less the mean or median (measures of center)? When we add the 15th car which changes les the standard deviation or the IQR (measures of dispersion)? 3-18

19 Suppose a 15 th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range? Without 15 th car With 15 th car Mean 32.1 mph 36.7 mph Median 32.5 mph 33 mph Standard deviation 6.2 mph 18.5 mph IQR 10 mph 11 mph 3-19

20 Outliers

21 EXAMPLE Determining and Interpreting the Interquartile Range Check the speed data for outliers. Step 1: The first and third quartiles are Q 1 = 28 mph and Q 3 = 38 mph. Step 2: The interquartile range is 10 mph. Step 3: The fences are Lower Fence = Q 1 1.5(IQR) Upper Fence = Q (IQR) = (10) = (10) = 13 mph = 53 mph Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers. 3-21

22 Sample Problem For the following data of rainfall for Chicago, IL determine the following: 1. The Quartiles & Median 2. The IQR 3. Determine if there are any outliers

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