stat-60_lect-05_April_08

stat-60_lect-05_April_08 - STATS 60 Spring 2008 April 8...

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STATS 60, Spring 2008 April 8, Lecture 05 1 Freedman Chapters 5 & 6 Normal curve The normal curve ( normal probability density function , or normal pdf ) can be used to describe the distribution of a wide range of populations and samples, and it plays a large role in many aspects of statistics. To begin our discussion of the normal pdf, look at a standard normal curve , which has a mean of zero and variance of one, and ask yourself, "What four things stand out most?" We will examine the formula for the normal curve, but lets start by noting that we often would like to express a sample observation x as being so many standard deviations away from the mean. The formal way to do that is through the z -score: z -score = x - ¯ x s . The corresponding z -score for a population observation would be x - μ σ . Another way to say this is that the z -score measures the distance from the mean in units of standard deviation. Freedman calls the z -score the number of standard units from the mean. As an example, say you have three sample observations of 2,5, and 12. In R you could calculate the z -score by: > dat = c(2,5,12) > m = mean(dat) > s = sd(dat) > z = (dat - m) / s > z [1] -0.8444407 -0.2598279 1.1042687 We see that the z -score for the value 12 would be 1.10, for example. Why might the z -score be a more useful measure than the distance from the mean itself? Now, when working with a pdf, we are more concerned with the area under the curve than with values on the y -axis. We know that the y -axis is adjusted so that the total area under the density curve is equal to one. Let’s consider a standard normal pdf, where the mean is zero and the standard deviation is one. Of course, the area under the curve is also one. The curve is symmetric, so we know that at x = 0 half of the area is to the left. We also can intuit that if we picked a random number from the distribution, there would be a 50 percent chance that it would be lower than zero. This simple example demonstrates how probability is related to the density curve...or rather, how probability is related to the area under the density curve. We pick any value x , and the area under the curve to the left of x tells us the probability that a random selection from the distribution will be less than or equal to x . So for example we could ask, "What is the probability that x is less than - 2 . Look at the standard normal curve and take a guess.
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