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Z scores MM3D3

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**Recall: Empirical Rule**

68% of the data is within one standard deviation of the mean 95% of the data is within two standard deviations of the mean 99.7% of the data is within three standard deviations of the mean 99.7% 95% 68% 𝑥 −3𝑠 𝑥 −2𝑠 𝑥 −𝑠 𝑥 𝑥 +𝑠 𝑥 +2𝑠 𝑥 +3𝑠

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**Example IQ Scores are Normally Distributed with N(110, 25)**

Complete the axis for the curve 99.7% 95% 68% 35 60 85 110 135 160 185

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**Example What percent of the population scores lower than 85? 16% 99.7%**

95% 68% 35 60 85 110 135 160 185

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**Example What percent of the population scores lower than 100? 99.7%**

95% 68% 35 60 85 100 110 135 160 185

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Z Scores Allow you to get percentages that don’t fall on the boundaries for the empirical rule Convert observations (x’s) into standardized scores (z’s) using the formula: 𝑧= 𝑥−𝜇 𝜎

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**Practice: Convert the following IQ Score N(110, 25) to z scores:**

100 125 75 140 45 -.4 .6 -1.4 1.2 -2.6

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Z Scores The z score tells you how many standard deviations the x value is from the mean The axis for the Standard Normal Curve: -3 -2 -1 1 2 3

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Z Score Table: The table will tell you the proportion of the population that falls BELOW a given z-score. The left column gives the ones and tenths place The top row gives the hundredths place What percent of the population is below .56? .7123 or 71.23%

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Z Score Table: The table will tell you the proportion of the population that falls BELOW a given z-score. The left column gives the ones and tenths place The top row gives the hundredths place What percent of the population is below .4? .6554 or 65.54%

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Practice: Use your z score table to find the percent of the population that fall below the following z scores: z < 2.01 z < 3.39 z < 0.08 z < -1.53 z < -3.47 97.78% 99.97% 53.19% 6.30% .03%

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Using the z score table You can also find the proportion that is above a z score Subtract the table value from 1 or 100% Find the percent of the population that is above a z score of 2.59 z > 2.59 .0048 or .48% Find the percent of the population that is above a z score of -1.91 z > -1.91 .9719 or 97.19%

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Using the z score table You can also find the proportion that is between two z scores Subtract the table values from each other Find the percent of the population that is between .27 and 1.34 .27 < z < 1.34 .3035 or 30.35% Find the percent of the population that is between and 1.89 -2.01 < z < 1.89 .9484 or 94.84%

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Practice worksheet

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**Application 1 IQ Scores are Normally Distributed with N(110, 25)**

What percent of the population scores below 100? Convert the x value to a z score 𝑧= 𝑥−𝜇 𝜎 z < -.4 Use the z score table .3446 or 34.46% = 100−110 25 =−.4

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**Application 2 IQ Scores are Normally Distributed with N(110, 25)**

What percent of the population scores above 115? Convert the x value to a z score 𝑧= 𝑥−𝜇 𝜎 z > .2 Use the z score table .5793 fall below .4207 or 42.07% = 115−110 25 =.2

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**Application 3 IQ Scores are Normally Distributed with N(110, 25)**

What percent of the population score between 50 and 150? Convert the x values to z scores 𝑧= 𝑥−𝜇 𝜎 -2.4 < z < 1.6 Use the z score table .9452 and .0082 This question is asking for between, so you have to subtract from each other. .9370 or 93.7% = 150−110 25 =1.6 = 50−110 25 =−2.4

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Practice worksheet

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