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Warm Up 9/13 and 9/14 From Chapter 2, Complete the following exercises: Do #19 Do # 21 Do #33 Show all work including formula’s and calculations

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Standardizing and Z- scores If x is an observation form a distribution that has a mean and standard deviation , the standardized value of x is a standardized value that is often called a z-score. z-score – tells how many standard deviations the original observation falls from the mean, and in which direction. Standardizing normal distributions make them all the same. They are still normal. It produces a new variable that has the standard normal distribution.

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Standard Normal Distribution N(0,1) Mean = 0, standard deviation = 1. If a variable x has any normal distribution N( , ), then the standardized variable has the standard normal distribution. When a score is expressed in standard deviation units, it is referred to as a Z-score A score that is one standard deviation above the mean has a Z-score of 1. A score that is one standard deviation below the mean has a Z-score of -1. A score that is at the mean would have a Z-score of 0. The normal curve with Z-scores along the x-axis looks exactly like the normal curve with standard deviation units along the x-axis.

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Normal Distribution Calculations Since all normal distributions are normal when we standardize, we can find the areas under any normal curve from a single table. Table A (inside the front cover of text) gives areas under the curve for standard normal distribution.

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To find normal proportions given observed variable: State problem, draw sketch. Standardize x using z =(x- )/ . Use the table. (Note if you are finding area to left or area to right.) OR Use the calculator NormCfd(lower z, upper z) = area Write conclusion in context of problem. A z-score outside the range of table A can be considered essentially 0 since the area of the last entry is.002. Anything beyond that will continue to decrease and is insignificant BUT NOT ZERO (close to zero, essentially zero, but not equal to zero). SHOW YOUR WORK!!

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Example The distribution of mathematics SAT scores is approximately normal with a mean of 500 points and a standard deviation of 100. Eleanor scores a 680 on this section. How many standard deviations away from the mean is she? In what direction? What is her percentile?

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To find normal areas using calculator: Press 2nd VARS (DISTR) Choose the 2nd option normalcdf( Complete the command normalcdf(lower z, upper z, , ) For infinity, use 1EE99 EE is above the comma. To type EE, type 2 nd comma

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To find x-value given normal proportion State problem, draw sketch. Use table. (Locate entry closest to given % and find corresponding z-score.) OR Use Calculator Function InvNorm(% written as a decimal) = z score Un-standardize the value using z =(x- )/ . Write conclusion in context of problem. SHOW YOUR WORK!!

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Eleanor wants to be in the 98 th percentile. How many standard deviations above the mean is this? What SAT score does she need to achieve her goal?

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InvNorm(%/100) = z!! Type 2 nd Vars Go to 3: InvNorm( Type in the percentile (% below) as a decimal This is the z-score associated with this percentile.

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ShadeNorm(lower, upper, mu, sigma) The following density curve was formed by using the ages (in months and years) of a AP Stats class. It shows the “area” or percentage of students above the age of 17. 23.6479 represents 10 above the mean. The interval is theoretically [17, ]. 10 standard deviations above the mean of a normal curve is as accurate as needed. Notice the area is.591282. The mean was 17.15, so you would expect the area to be over half.

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So, what if you have some data… And, you want to know if it is essentially normally distributed… 1. WHY WOULD WE WANT TO KNOW THAT?? Because if it is, then we can approximate proportions between values using the normal table or normal calculator function! 2. Okay, maybe that might be useful. So how do we decide?

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Normal Probability Plot!! What? A plot made by looking at how data points are spread out… do they pretty much follow the 68-95-99.7 rule? If so, your Normal Probability Plot should look LINEAR. If not approximately normally distributed it won’t look linear. How? Enter data in L1 In StatPlot choose the very last picture– the one that looks like a line, tell it where your data is stored (ie L1), choose ‘X’ and any mark you would like. Graph with ZoomStat.

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Other ways to decide on normality… See if it follows the 68-95-99.7 rule using the standard deviation and the mean If there are 100 data points, the 34 in either direction from the median should be between and +/- (same for 43 +/- 2 and 49 +/- 3 ) Kind of tedious Look at the data to see if it is roughly symmetric (box plot, histogram) Doesn’t give you an idea of whether it follows the Empirical Rule Moral of the story… use the Normal Probability Plot but be aware that there are other ways to check for normality.

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Assignment: HW-Finish Reading and taking notes Chapter 2 Do #’s 30, 31, 33, 35, 38, 40, 44, 45, 47, 49 Show all calculations and make sure you write your answers contextually with meaning. Fill in toolkit and study for TOOLKIT quiz. Sept 19th Aday and Sept 20 th Bday

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