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ALGEBRA II WEDNESDAY, JANUARY 28, 2015 B DAY DRILL: How many students will be taking the SAT or have taken it? What is the scoring range ? What do you think the mean score is for math in 2012? The standard deviation was 114 in 2012. What does this mean? What would a lower SD mean? What would a higher SD mean?

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UNIT IV: Inferences and Conclusions from Data TOPIC: Normal Distributions, Z-Scores, Population Percentages Students will be able to… – Summarize, represent, and interpret data on a single count or measurement variable. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Standards for Mathematical Practices Used: MP1: Make sense of problems and persevere in solving them. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure.

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What should be taken into consideration when evaluating a statistical model? 1.How well the structural part of the model describes the data 2.How useful the model is Univariate Data: values that can be modeled as varying about a single numeric value Structure: single value (such as mean) Variability: deviations from the structure/central location

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Reviewing Distribution Types Sample Distribution Sample Mean: Sample Standard Deviation: S Population Distribution- when you only care about data you have Population Mean: Population Standard Deviation: Sampling Distribution Sampling Mean: Sampling Standard Deviation:

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Review of Standard Deviation (Algebra I) Standard deviation gives an indication of how much, on average, data values deviate from the structural part of the model (typically the mean). Sample Standard Deviation: Population Standard Deviation:

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Standard Deviation

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Mr. Notachance is marking a test. Here are the students results (out of 60 points): 20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17 Most students did not even get 30 out of 60, and most will fail. Mr. Notachance decides to standardize all the scores and only fail people 1 standard deviation below the mean. Which scores will be considered failing? S.ID.A.4

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Mr. Notachance is marking a test. Here are the students results (out of 60 points): 20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17 Mean = 23 Population Standard Deviation = 6.6 Anyone with a 16.4 or higher will pass the test S.ID.A.4

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Normal Distribution S.ID.A.4

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Sample IQ Scores 125, 71, 138, 96, 105, 111, 119, 89, 116, 84, 98, 90, 110, 92, 86, 85, 122, 87, 62, 78, 99, 94, 132, 116, 99, 109, 76, 91, 65

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Sample ACT Scores 21, 23, 23, 23, 24, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 28, 28, 29, 29, 29, 30, 31, 31, 32, 32, 36

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S.ID.A.4

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Standard Scores The number of standard deviations from the mean is also called the “standard score” or “z-score” To convert a value (x) to a Standard Score (z-score) 1.Subtract the mean 2.Divide by the standard deviation S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. You score a 600 on one of the sections. How do you compare to other students taking the SAT? S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. You score a 600 on one of the sections. How do you compare to other students taking the SAT? z-score = 1 S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. You score a 600 on one of the sections. How do you compare to other students taking the SAT? z-score = 1 68% of students fall within 1 z-score of the mean 32% of students have scores more than 1 standard deviation from the mean Half of those students (16%) would have scored better than you. S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. Your best friend scores a 680 on one of the sections. How does your friend compare to other students taking the SAT? S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. Your best friend scores a 680 on one of the sections. How does your friend compare to other students taking the SAT? z-score = 1.8 Need to find the normal percentile S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. Your best friend scores a 680 on one of the sections. How does your friend compare to other students taking the SAT? z-score = 1.8 Need to find the normal percentile 96.4% is a cumulative percentile, so 96.4% scored 680 or lower. So, 3.6% would have scored better than your friend. S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. Your brother scores a 440 on one of the sections. How does your friend compare to other students taking the SAT? S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. Your brother scores a 440 on one of the sections. How does your friend compare to other students taking the SAT? z-score = -0.6 Need to find the normal percentile S.ID.A.4

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Normal Percentiles SAT scores fall within a normal distribution. The mean SAT score for an individual section is 500, with a standard deviation of 100. Your brother scores a 440 on one of the sections. How does your friend compare to other students taking the SAT? z-score = -0.6 Need to find the normal percentile 27.43% is a cumulative percentile, so 27.43% scored 440 or lower. So, 72.57% would have scored better than your friend. S.ID.A.4

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Normal Percentiles There was a study called the HANES study which looked at physical characteristics of Americans. In the sample, there were 6,588 women age 18-74. Their average height was 63.5 inches, and the standard deviation of height was 2.5 inches. If a woman is 68.9 inches tall, how does her height compare to heights of other American women?

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Z-score = 2.16

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I am taller than 98.46% of American women.

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PARCC Samples

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