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3/19/2013 7.2: Standard Units and Areas Under the Standard Normal Distribution Standard score - used to standardize the distributions so that we can use one table of areas for all normal distributions. Z score - gives the number of standard deviations between the original measurement x and the mean of the x distribution. z x Raw score - given an x distribution with mean and standard deviation, the raw score x corresponding to a z score is x z Ex: A student has computed that it takes an average of 35 minutes with a standard deviation of 3 minutes to get ready for school in the morning. a) One morning, it took the student 30 minutes to get ready. How many standard deviations from the mean is that? b) Is the z value positive or negative? Explain why. 1 3/19/2013 Ex: Our data is normally distributed with a mean of 20.8 and standard deviation of 5.5. Convert each of the following x intervals to z intervals a) x >21 b) 19 < x < 25 Convert each of the following z intervals to x intervals a) z < 1.45 b) -2.05 < z < -.8 Standard Normal Distribution - a normal distribution with mean 0 and standard deviation 1. Areas under the standard normal curve - each area is equal to the probability that the measurement of an item selected at random falls in this interval Diagram: 2 3/19/2013 Using a Standard Normal Distribution Table • left-tail style table 1. For areas to the left of a specified z value, use the table entry directly 2. For areas to the right of a specified z value, look up the table entry for z and subtract the area from 1. 3. For areas between two z values z1 and z2 (where z2 > z1), subtract the table area for z1 from the table area for z2. Ex: Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas a) To the left of z = .72 b) To the right of z = 1.45 c) Between z = -2.1 and z = 1.38 Ex: Find the indicated probability, and shade the corresponding area under the standard normal curve. a) P(z < -.23) b) P(z > -1.4) c) P(-.45 < z < 0) 3 3/19/2013 Assignment: p. 266-268 #1-8 all; #10-48 every 4th 4