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Applied Math 40S Slides Mar 16, 2007

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Normal Distributions, the Standard Normal Curve, z-scores and applications.

Normal Distributions, the Standard Normal Curve, z-scores and applications.

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    Applied Math 40S Slides Mar 16, 2007 Applied Math 40S Slides Mar 16, 2007 Presentation Transcript

    • Let's warm up with what we've already learned ... The frequency distribution table at right mark interval mark # of students 29 to 37 33 1 shows the midterm marks of 85 Senior 4 38 to 46 42 4 math students at Parksville High. The first 47 to 55 51 12 column shows the mark interval, the 56 to 64 60 18 second column the average mark within 65 to 73 69 24 each mark interval, and the third column 74 to 82 78 16 the number of students at each mark. 83 to 91 87 7 Calculate the mean math mark and the 92 to 100 96 3 standard deviation. Total 85 Calculate the number of students that have marks within one standard deviation of the mean. What percent of students have marks within one standard deviation of the mean? Construct both a frequency and a probability distribution for this data set.
    • Calculate the mean math mark and the mark interval mark # of students 29 to 37 33 1 standard deviation. 38 to 46 42 4 47 to 55 51 12 56 to 64 60 18 65 to 73 69 24 74 to 82 78 16 Calculate the number of students that have 83 to 91 87 7 marks within one standard deviation of the 92 to 100 96 3 mean. Total 85 What percent of students have marks within one standard deviation of the mean?
    • Construct both a frequency and a probability mark interval mark # of students 29 to 37 33 1 distribution for this data set. 38 to 46 42 4 47 to 55 51 12 56 to 64 60 18 65 to 73 69 24 74 to 82 78 16 83 to 91 87 7 92 to 100 96 3 Total 85
    • The Standard Normal curve - the standard normal curve is used to make comparisons between different normal distributions. This is done using z-scores which allows us to find a particular value , x , which lies on the standard normal curve. This is what it meant by "standardizing" the scores for a particular normal distribution. DICTIONARY
    • The Standard Normal curve - the standard normal curve is used to make comparisons between different normal distributions. This is done using z-scores which allows us to find a particular value , x , which lies on the standard normal curve. This is what it meant by "standardizing" the scores for a particular normal distribution. DICTIONARY
    • Properties of a normal distributions • Each value of mean and standard deviation determines a different normal distributions. • All normal distributions are symetrical about the mean. • 99.7% of all the data lies within approximately 3 standard deviations of the mean. • The area under the curve always equals one. • The x-axis is an asymptote for the curve. DICTIONARY Interactivate Normal Distribution
    • The 68-95-99 Rule DICTIONARY Generally speaking, approximately: • 68% of all the data in a normal distribution lie within the 1 standard deviation of the mean, • 95% of all the data lie within 2 standard deviations of the mean, and • 99.7% of all the data lie within standard deviations of the mean.
    • The contents in the cans of several cases of soft drinks were tested. The mean contents per can is 356 mL, and the standard deviation is 1.5 mL. (a) Two cans were randomly selected and tested. One can held 358 mL, and the other can 352 mL. Calculate the z-score of each. . (b) Two other cans had z-scores of -3 and 1.85. How many mL did each contain?
    • For a given set of data, µ = 23 and σ = 4.5. (a) What are µ and σ if 10 is added to each number in the set of data? (b) What are µ and σ if each number in the set of data is multiplied by 2?