CHAPTER 9

CHAPTER 9 - CHAPTER 9 NORMAL DISTRIBUTION AND SAMPLING...

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Unformatted text preview: CHAPTER 9: NORMAL DISTRIBUTION AND SAMPLING DISTRIBUTIONS 9.2 The Normal Distribution Most important probability distribution in statistics A sample statistic’s distribution will look like the normal distribution as n (n=number of observations in each random sample) increases Abraham de Moivre discovered the function rule for finding the height of the normal distribution, f(X), for any value of the random variable X Characteristics of the Normal Distribution If the probability distribution of a random variable X is given by the function rule for the normal distribution, then X is normally distributed Function rule for the normal distribution : f(X): height of the distribution at X μ: mean of a particular normal distribution : standard deviation of a particular normal distribution σ To find the size of areas under the distribution between values of X, use Appendix Table D.2 The normal distribution is: bell-shaped unimodal symmetric (mean, median, mode have the same value which is the highest point on the curve) Inflection points (where the curve changes from concave to convex & vv) of the curve are at μ+ and μ- σ σ Tails of the curve extend indefinitely in both directions. They get really close to the horizontal axis but never touch it. Total area under the curve = 1 Converting Scores to Standard Scores There are an infinite number of normal distributions since there are an infinite number of possible values for μ & so statisticians chose 1 normal distribution to be the standard. σ This means that we have to standardize all of the other normal distributions to this one and allows us to have 1 table for the normal distribution instead of an infinite number of tables. S tandard normal distribution – name for the normal distribution that has a mean of 0 and standard deviation of 1 (μ=0 & =1). This is the one statisticians chose to be the σ standard and the areas for this distribution are in Appendix Table D.2. Standard scores – denoted by z and represent the values of the random variable for this distribution Usually the random variable we are working with does not have a mean of 0 and standard deviation of 1 so we can not use Appendix Table D.2 to find the areas for the distribution. To use this table, all we have to do is transform the values for our random variable into standard scores. The formula we use to do this is: X: value of random variable : mean of the sample S: standard deviation of the sample When we apply this z-score transformation to each X in our distribution, the distribution of transformed scores will have a mean of 0 and standard deviation of 1 which is what we need to use the table for the standard normal distribution The z-score transformation alters the mean and standard deviation but not the relative location of scores in the distribution : mean of the z scores : standard deviation of the z scores When you graph the distribution of the X scores and the distribution of the z scores (the z scores are our X scores after the transformation) their shapes are identical but the central tendency and dispersion will be different. The shape of the distribution...
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CHAPTER 9 - CHAPTER 9 NORMAL DISTRIBUTION AND SAMPLING...

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