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Z-Scores 10/13/2015 Statistics Mr. DeOms
SWBAT analyze and calculate the z-scores from normal distribution data. SWBAT build community and understanding by sharing images of self.
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Frequency Distributions
Data Displays F - Box and Whiskers Quiz HW.13 Due Standard Deviation T - Z-Scores Th - Excel Skills M β Review Problem Set 2.2 Due W - Unit 2 Test
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π=π π=1
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For All Normal Distributions
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Example A If the diameter of a basketball is normally distributed, with a meanΒ (ΞΌ)Β of 9β³, and a standard deviationΒ (Ο)Β of 0.5β³, what is the probability that a randomly chosen basketball will have a diameter between 9.5β³ and 10.5β³? ππ.ππ% βππ.ππ% ππ.ππ% ππ% ππ.ππ% π" π.π" 10" 10.5"
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Example B If the height of women in the U.S. is normally distributed with ΞΌ=5β²8β²β² and Ο=1.5β²β², what is the probability that a randomly chosen woman in the U.S. is shorter than 5β²5β³? πππ.π% βππ.π% π.π% ππ% ππ.π% 5βπ" π β² π.π" πβ²π"
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Quick Practice 1. A normally distributed data set has π=10 and π=2.5, what is the probability of randomly selecting a value greater than 17.5 from the set? 2. A normally distributed data set has π=.05 and π=.01, what is the probability of randomly choosing a value between .05 and .07 from the set? 3. A normally distributed data set has π=514 and an unknown standard deviation, what is the probability that a randomly selected value will be less than 514?
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Soβ¦ for the first three standard deviations from the mean we know how to find the percentage of data included! If we want to know the percentage of the population above, below, or between any values β¦ we can use the z-score! The z-score tells us the number of standard deviations between any value and the mean of the population.Β
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Standard Deviation of population
Mean of population value π§= π₯βπ π z-score Standard Deviation of population
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Example C What is theΒ z-score of a value of 27, given a set mean of 24, and a standard deviation of 2? π= πβπ π π= (ππ)β(ππ) (π) π= π π π=π.π
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Example D What is the z-score of a value of 104.5, in a set with π=125 and π=6.2? π= πβπ π π= (πππ.π)β(πππ) (π.π) π= βππ.π π.π π=βπ.ππ
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Quick Practice 4. What is theΒ z-score of the price of a pair of skis that cost $247, if the mean ski price is $279, with a standard deviation of $16? 5. What is theΒ z-score of a 5-scoop ice cream cone if the mean number of scoops is 3, with a standard deviation of 1 scoop? 6. What is theΒ z-score of the weight of a cow that tips the scales at 825 lbs, if the mean weight for cows of her type is 1150 lbs, with a standard deviation of 77 lbs?
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Tips for reading Z-Score Tables
Z-score tables give us the probability that a value or any value less than it, will occur. To find the percentage of valuesΒ greaterΒ than aΒ z-score, subtract the chart value from 1.
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ππ.ππ% What is the probability of a z-score less than 0.54? ____________ What is the probability of a z-score greater than 0.54? ____________ What is the probability of a z-score less than -0.93? ____________ What is the probability of a z-score greater than -0.93? ____________ What is the probability of a z-score greater than 0.81? ____________ What is the probability of a z-score less than -0.81? ____________ ππ.ππ% ππ.ππ% ππ.ππ% ππ.ππ% ππ.ππ%
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Examples E. What is the probability that a value with a z-score less than 2.47 will occur in a normal distribution? F. What is the probability that a value with a z-score greater than 1.53 will occur in a normal distribution? G. What is the probability of a random selection being less than 3.65, given a normal distribution with π=5 πππ π=2.2?
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Homework Due Monday: Problem Set 2.2 Unit Exam Wednesday (10/21)
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