Applied Regression Modeling. Iain Pardoe

Applied Regression Modeling - Iain Pardoe


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alt="images"/> is normal with mean 280 and standard deviation images. Then the standardized images‐value from images,

equation

      is standard normal with mean 0 and standard deviation 1. From the normal table in Section 1.2, the 90th percentile of a standard normal random variable is 1.282 (since the horizontal axis value of 1.282 corresponds to an upper‐tail area of 0.1). Then

equation

      Thus, the 90th percentile of the sampling distribution of images is images (to the nearest images). In other words, under repeated sampling, images has a distribution with an area of 0.90 to the left of images (and an area of 0.10 to the right of images). This illustrates a crucial distinction between the distribution of population images‐values and the sampling distribution of images—the latter is much less spread out. For example, suppose for the sake of argument that the population distribution of images is normal (although this is not actually required for the central limit theorem to work). Then we can do a similar calculation to the one above to find the 90th percentile of this distribution (normal with mean 280 and standard deviation 50). In particular,

equation Graph depicts the central limit theorem in action. The upper density curve (a) shows a normal population distribution for Y with mean 280 and standard deviation 50: the shaded area is 0.10, which lies to the right of the 90th percentile, 344.100. The lower density curve (b) shows a normal sampling distribution for MY with mean 280 and standard deviation 9.129: the shaded area is also 0.10, which lies to the right of the 90th percentile, 291.703. It is not necessary for the population distribution of Y to be normal for the central limit theorem to work—we have used a normal population distribution here just for the sake of illustration. equation

      So, the probability that images is greater than 291.703 is 0.10.

      1.4.2 Central limit theorem—t‐version

      For example, the following table shows critical values (i.e., horizontal axis values or percentiles) and tail areas for a t‐distribution with 29 degrees of freedom: Probabilities (tail areas) and percentiles (critical values) for a t‐distribution with images degrees of freedom.

Upper‐tail area 0.1 0.05
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