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Aim: How do we apply the characteristics of normal distribution?. 10 coins tossed 100 times result in the following table. Draw a histogram based on the table and determine the mean, x. 1. 3. 5. 7. 9. Do Now:. Normal Curve – the ‘Bell Curve’. also mode & median. symmetrical.

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## Aim: How do we apply the characteristics of normal distribution?

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**Aim: How do we apply the characteristics of normal**distribution? 10 coins tossed 100 times result in the following table. Draw a histogram based on the table and determine the mean, x. 1 3 5 7 9 Do Now:**Normal Curve – the ‘Bell Curve’**also mode & median symmetrical The most prominent probability distribution in statistics.**99.5% of data values**95% of data values 68% of data values 13.5% 13.5% Normal Distribution 34% 34% 68% of data lie within 1 standard deviation of mean. 95% of data within 2 standard deviations of mean. 99.5% of data within 3 standard deviations of mean.**99.5% of data values**95% of data values 68% of data values Percentile 13.5% 34% 34% 13.5% 2.5 16 50 84 97.5 percentile of a score or a measure indicates what percent of the total frequency scored at or below that measure.**128**133 143 148 Model Problem • In a normal distribution, the mean height of 10-year-old children is 138 centimeters and the standard deviation is 5 centimeters. Find the heights that are • exactly one standard deviation above and below the mean • two standard deviations above and below the mean • In a normal distribution, the mean height of 10-year-old children is 138 centimeters and the standard deviation is 5 centimeters. Find the heights that are • exactly one standard deviation above and below the mean • two standard deviations above and below the mean**68%**95% 10-year-old Model Problem In a normal distribution, the mean height of 10-year-old children is 138 centimeters and the standard deviation is 5 centimeters. 13.5% 13.5% 34% 34% 128 133 143 148 2.5 16 50 84 97.5 Of the children: 68% are between 133 and 143 centimeters tall 95% are between 128 and 148 centimeters tall 34% are between 138 and 142 centimeters tall**13.5%**34% 34% 13.5% 10-year-old Model Problem In a normal distribution, the mean height of 10-year-old children is 138 centimeters and the standard deviation is 5 centimeters. 128 133 143 148 2.5 16 50 84 97.5 A ten-year-old who is 133 cm. tall is at the 16th percentile; 16% are shorter, 84% taller Heights that would occur less than 5% of the time: heights of less than 128 cm. or more than 148 cm.**2pt. Regents Question**Assume that the ages of first-year college students are normally distributed with a mean of 19 years and standard deviation of 1 year. To the nearest integer, find the percentage of first-year college students who are between the ages of 18 years and 20 years inclusive. To the nearest integer, find the percentage of first-year college students who are 20 years or older.**13.5%**34% 34% 13.5% 34 43 61 70 2.5 16 50 84 97.5 Model Problem • Scores on the Preliminary Scholastic Aptitude Test (PSAT) range from 20 to 80. For a certain population of students, the mean is 52 and the standard deviation is 9. • A score at the 65th percentile might be • 49 2) 56 3) 64 4) 65 • Which of the following scores can be expected to occur less than 3% of the time? • 39 2) 47 3) 65 4) 71**13.5%**34% 34% 13.5% 12 40 2.5 16 50 84 97.5 Model Problem In the diagram, the shaded area represents approximately 68% of the scores in a normal distribution. If the scores range from 12 to 40 in this interval, find the standard deviation.**4pt. Regents Question**Twenty high school students took an examination and received the following scores: 70, 60, 75, 68, 85, 86, 78, 72, 82, 88, 88, 73, 74, 79, 86, 82, 90, 92, 93, 73 Determine what percent of the student scored within one standard deviation of the mean. Do the results of the examination approximate a normal distribution? Justify your answer.**Model Problem**In 2000, over 1.2 million students across the country took college entrance exams. The average score on the verbal section showed no improvement over the average scores of the previous 4 years. The average score on the mathematics section was 3 points higher than the previous year’s average. What is the probability that a student’s verbal score is from 401 to 514?**Model Problem**In 2000, over 1.2 million students across the country took college entrance exams. The average score on the verbal section showed no improvement over the average scores of the previous 4 years. The average score on the mathematics section was 3 points higher than the previous year’s average. What is the probability that a student’s math score is greater than 727?**Model Problem**In 2000, over 1.2 million students across the country took college entrance exams. The average score on the verbal section showed no improvement over the average scores of the previous 4 years. The average score on the mathematics section was 3 points higher than the previous year’s average. Both Susanna’s math and verbal scores were more than one standard deviation above the mean, but less than 2 standard deviations above the mean. What are the lower and upper limits of Susanna’s combined score?

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