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THE STANDARD NORMAL DISTRIBUTION Individual normal distributions all have means and standard deviations that relate to them specifically. In order to compare two unlike distributions, it is necessary to standardize them. This allows us to compare values of identical proportions against different scales. Example: Elmer scored 680 on the math portion of the SAT [ N(500, 100)]. Otis scored 29 on the ACT mathematics test [ N(18, 6)]. Who had the better score?

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What we need to know is exactly how many standard deviations each score is from its respective mean. In other words we need find the correct distance of each boy’s score from the mean of his distribution. We do this by setting each distribution mean to 0 and then measuring the number of standard distributions from 0 to that person’s score. This is called standardizing the value. Subtract the mean from the value of interest and divide by the standard deviation. This new value is called a “z” score.

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In a standardized distribution, the mean is 0 and the standard deviation is 1. The distance from the mean is a count of the number of standard deviations a value is from the mean.

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STANDARD NORMAL DISTRIBUTION PROBLEMS The heights of women ages 20 to 29 are N(64, 2.7). Men the same age are N(69.3, 2.8). What are the z-scores for a woman 6 feet tall and a man 6 feet tall? What information do the z- scores give that the actual heights do not? Find the proportion of values from a standard normal distribution that satisfies each of the following. In each case, sketch a normal curve and shade the area under the curve that is the answer to the question. a)Z ≤ 1.85b) z > 1.85 c) Z < - 0.66d) -0.66 < z < 1.85

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Find the z-value of a standard normal variable that satisfies each of the following conditions: a) 20% of the observations fall below z. b) 30% of the observations fall above z. The Wechsler Adult Intelligence Scale (WAIS) is the most common “IQ Test”. The scale is approximately normal with mean 100 and standard deviation 15. People with WAIS scores below 70 are considered mentally challenged when, for example, applying for Social Security disability benefits. What percent of adults are disabled by this definition?

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Z-SCORES AND THE EMPIRICAL RULE Use the table of standard normal probabilities to find the proportion of the normal curve that falls within one standard deviation of its mean. Repeat for two and three standard deviations. Determine the z-score such that the area under a normal curve between it and its negative is.5000. Then subtract these two z-scores to find the interquartile range of a normal distribution. Recall that an outlier is any value falling more than 1.5(IQR) away from its closest quartile. Determine the z-scores for outliers and the proportion of values in a normal distribution that are considered outliers.

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