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Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 113

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5.2 Properties of Normal Distribution Is it Normal? Circumference of 10,000 adult wrists… YES--NORMAL High jump results for all High School track meets in America… YES--NORMAL Calories consumed by Sumo wrestlers compared to calories consumed by the general public… SOMEWHAT NORMAL—SKEWED RIGHT COMPARED TO GENERAL PUBLIC Prices of a particular Video game a different retail stores… YES--NORMAL Weights of 50 randomly selected baseballs… YES--NORMAL

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5.2 Properties of Normal Distribution Remember what makes a variable normally distributed ? 1. The variable is determined by many different factors (i.e. physiological variables or performance variables) 2. The frequencies of the variable cluster around a single peak that is near the mean of the distribution. 3. The frequencies of the variable are symmetric about the peak. 4. Large deviations from the mean are increasingly rare.

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5.2 Properties of Normal Distribution THE EMPIRICAL RULE Data sets having a normal, bell-shaped distribution, the following properties apply: http://www.stat.tamu.edu/~west/applets/empiricalrule.html

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5.2 Properties of Normal Distribution 68-95-99.7 Rule of Normal Distribution 68 % of the normal distribution occurs within 1 standard of deviation from the mean 95 % of the normal distribution occurs within 2 standards of deviation from the mean 99.7 % of the normal distribution occurs within 3 standards of deviation from the mean

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5.2 Properties of Normal Distribution Using 68-95-99.7 Rule for Normal Distribution: If there are 600 total values. Approximately, how many values occur within 1 standard of deviation from the mean? 2 standards of deviation? 3 standards of deviation? 408, 570, and 598.2

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5.2 Properties of Normal Distribution Using 68-95-99.7 Rule for Normal Distribution: If there are 325 values within 1 standard of deviation from the mean, approximately, how many values exist in the data set? Approximately, how many values occur within 3 standards of deviation from the mean? 478, and 477

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5.2 Properties of Normal Distribution Unusual Values… Values that are more than 2 standards of deviations away from the mean.

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5.2 Properties of Normal Distribution EXAMPLE: Suppose you measure your heart rate at noon every day for a year and record the data. You discover that the data has a normal distribution with a mean of 66 and a standard of deviation of 4. On how many days was your heart rate below 58 beats per minute? A heart rate of 58 is 8 below the mean (2 standards of deviation below). From the 68-95-99.7 rule, 95% of the data occurs 2 standards of deviation from the mean. If we want to know how many days our heart rate was below 58 this would imply 2.5% of our data, because ((100%-95%)÷2=2.5%). Therefore, on 2.5% of 365 days in a year your heart rate is below 58 or roughly 9 days.

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5.2 Properties of Normal Distribution STANDARD SCORES (Z-scores) – The number of standard deviations between a particular data value and the mean is called its standard score, usually denoted z. Data values below the mean have a negative standard score, and data values above the mean have a positive standard score.

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5.2 Properties of Normal Distribution Z-SCORE: The Stanford-Binet IQ test is scaled so that scores have a mean of 100 and a standard deviation of 16. Find the z-scores for IQ’s of 85, 100, 125. 85 z-score = 100 z-score = 125 z-score =

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5.2 Properties of Normal Distribution Z-SCORES and Percentiles: TABLE 5.1 page 211. Standard score ≈ percentile i.e. Z-ScorePercentile ≈ -1.2 and -1.312 th ≈ -0.7 and -0.6525 th ≈ -0.35 and -0.3037 th ≈ 0.0 and 0.050 th ≈ 0.30 and 0.3562 nd ≈ 0.65 and 0.7075 th ≈ 1.1 and 1.287 th ≈ 3.5 and above99.98 th AreaOfANormalDistribution.nbp

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5.2 Properties of Normal Distribution MORE Z-SCORES… The heights of American women aged 18 to 24 are normally distributed with a mean of 65 inches and a standard deviation of 2.5 inches. In order to serve in the U.S. Army, women must be between 58 inches and 80 inches tall. What percentage of women are ineligible to serve based on their height? The z-score of 58 inches is -2.8 which corresponds to the 0.26 percentile from Table 5.1 on page 232. The z-score of 80 is 6.0 and corresponds to above the 99.98 percentile. Thus, 0.26% of all women are too short, and 0.02% of all women are too tall. Altogether, this means that 0.28% of all women are ineligible for the army based on their height. Hence, roughly one out of 358 women are ineligible.

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QUANITIFING THE POWER (Probability) OF NORMAL DISTRIBUTION AND STANDARD DEVIATION: μ ± 1σ → 0.682689492137 → 1 in 3 → weekly μ ± 2σ → 0.954499736104 → 1 in 22 → monthly μ ± 3σ → 0.997300203937 → 1 in 370 → yearly μ ± 4σ → 0.999936657516 → 1 in 15,787 → every 60 years (once in a lifetime) μ ± 5σ → 0.999999426697 → 1 in 1,744,278 → every 5,000 years (once in history??) μ ± 6σ → 0.999999998027 → 1 in 506,842,372 → every 1.5 million years (essentially never??) Thus for a daily process, a 6σ event is expected to happen less than once in a million years.

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5.2 Properties of Normal Distribution HOMEWORK: Pg 202 # 23-26 Pg 212 #1-11 Pg 213 #13-24 Pg 213 #37, 38, 42

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