3.5 Applying the Normal Distribution: Z-Scores How can the Normal Distribution be used to accurately determine the percentage of data that lies above or below a given data value? How can the Normal Distribution be used to compare data from two different data sets? Enter… “The Standard Normal Distribution”

3.5 Applying the Normal Distribution: Z-Scores Two students from different schools have a mark in a MDM 4U class. Adam has a mark of 83 Brenda has a mark of 84 Who has a better grade?

3.5 Applying the Normal Distribution: Z-Scores Not so easy to answer… Adam’s class average is 70 with a sdev of 9.8 Brenda’s class average is 74 with a sdev of 8 (Assume a Normal Distribution of marks in both classes) To make the comparison – must STANDARDIZE the distributions.

3.5 Applying the Normal Distribution:Z-Scores Standard Normal Distribution Special normal distribution with a mean of 0 and a standard deviation of 1 X~N(0,12) N(2.1, 9) ≡ mean of 2.1 and sdev of 3 Each element of a normal distribution can be translated to the same place on a Standard Normal Distribution by determining the number of standard deviations a given score lies away from the mean.

3.5 Applying the Normal Distribution:Z-Scores For a given score, x, we can say: ‘z’ is the number of sdev the score lies above or below the mean Solving for z: This is the z-score of a piece of data

3.5 Applying the Normal Distribution: Z-Scores A positive z-score indicates the value lies above the mean A negative z-score indicates the value lies below the mean

3.5 Applying the Normal Distribution: Z-Scores Calculating the z-scores for Adam & Brenda Now you can see that Adam has the better mark as he is 1.33 sdevs from the mean vs 1.25 sdevs in Brenda’s case.

3.5 Applying the Normal Distribution: Z-Scores Z-Score Table Used to find the proportion of data that has an equal or lesser score than a given value under the standardized normal distribution curve Found in back of textbook Pg 398 & 399

3.5 Z-Score Tables

3.5 Applying the Normal Distribution: Z-Scores Example The annual returns from a particular mutual fund are believed to be normally distributed. A sample of the last 15 years of historic returns are listed in the following table. Determine the mean & standard deviation of the annual return. What is the probability that an annual return will be: At least 9%? Negative? What is the probability the investment will yield a return greater than 6%?

3.5 Applying the Normal Distribution: Z-Scores Year Return (%) 1 7.2 6 19.3 11 6.4 2 12.3 7 12.2 12 27.0 3 17.1 8 -13.1 13 14.5 4 17.9 9 20.2 14 25.2 5 10.8 10 18.6 15 -0.5

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