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Z scores MM3D3.

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Presentation on theme: "Z scores MM3D3."โ€” Presentation transcript:

1 Z scores MM3D3

2 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% ๐‘ฅ โˆ’3๐‘  ๐‘ฅ โˆ’2๐‘  ๐‘ฅ โˆ’๐‘  ๐‘ฅ ๐‘ฅ +๐‘  ๐‘ฅ +2๐‘  ๐‘ฅ +3๐‘ 

3 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

4 Example What percent of the population scores lower than 85? 16% 99.7%
95% 68% 35 60 85 110 135 160 185

5 Example What percent of the population scores lower than 100? 99.7%
95% 68% 35 60 85 100 110 135 160 185

6 Z Scores Allow you to get percentages that donโ€™t fall on the boundaries for the empirical rule Convert observations (xโ€™s) into standardized scores (zโ€™s) using the formula: ๐‘ง= ๐‘ฅโˆ’๐œ‡ ๐œŽ

7 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

8 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

9 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%

10 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%

11 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%

12 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%

13 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%

14 Practice worksheet

15 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 ๐‘ง= ๐‘ฅโˆ’๐œ‡ ๐œŽ z < -.4 Use the z score table .3446 or 34.46% = 100โˆ’110 25 =โˆ’.4

16 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 ๐‘ง= ๐‘ฅโˆ’๐œ‡ ๐œŽ z > .2 Use the z score table .5793 fall below .4207 or 42.07% = 115โˆ’110 25 =.2

17 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 ๐‘ง= ๐‘ฅโˆ’๐œ‡ ๐œŽ -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% = 150โˆ’110 25 =1.6 = 50โˆ’110 25 =โˆ’2.4

18 Practice worksheet


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