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Section 2.2: Normal Distributions We learned in section 2.1 that some density curves are called normal curves. All normal curves are density curves, but not all density curves are normal. Normal distributions are sometimes called Gaussian because they were studied by Carl Friedrich Gauss. Empirical Rule: ex) Suppose math SAT scores follow a normal distribution with a mean of 450 and a standard deviation of 100. a. What scores mark the middle 68% of the data? b. middle 95% c. middle 99.7% d. What is the probability of scoring above 450? e. What is the probability of scoring above 650? f. What is the probability of scoring below 150? Now that we have learned that normal curves and the Empirical Rule can be used to approximate probabilities and area under the curve, we can move on to a more exact method of finding probability. All normal distributions are the same if they are measured in units of about the mean . Converting scores from their original values to standard deviation units is called standardizing. z-score - Standard Normal Distribution: The area under the curve is a proportion of the observations in the distribution. ex) Let's revisit SAT scores. Remember that the mean math SAT score is 450 and the standard deviation is 100. If you scored a 540, what percent scored below you? Now that you have a z-score, what do you do with it? ex) Suppose the length of a human pregnancy follows a normal distribution with a mean of 268 days andĪ = 15. A baby is considered premature if they are born 3 weeks early. What is the probability that a baby will be premature? ex) What is the probability a pregnancy will last more than 281 days? ex) What is the probability a pregnancy will last between 260 and 270 days? Sometimes you will be given an area or proportion or percentile and asked to find a value on the curve from that. ex) What math SAT score would you get if you scored in the 80th percentile? Assessing Normality: