Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner Figure 11.1 The law of large numbers in action: as we take more observations, the sample mean x always approaches the mean  of the population. ¯

Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner ¯ ¯ Figure 11.2 The idea of a sampling distribution: take many samples from the same population, collect the x’s from all the samples, and display the distribution of the x’s. The histogram shows the results of 1000 samples.

Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner Figure 11.3 The distribution of single observations (the population distribution) compared with the sampling distribution of the means x of 10 observations, for Example Both have the same mean, but averages are less variable than individual observations. ¯

Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner Figure 11.4 The central limit theorem in action, for Example (a) The distribution of earned income in a population of 75,310 households.

Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner Figure 11.4 (Continued) (b) The distribution of the mean earnings for 500 SRSs of 100 households each from this population.

Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner Figure 11.4 (Continued) (c) The distribution of the sample means in more detail: the shape is close to Normal.

Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner Figure 11.5 The central limit theorem in action, for Example The distribution of sample means x from a strongly non-Normal population becomes more Normal as the sample size increases. (a) The distribution of 1 observation. (b) The distribution of x for 2 observations. (c) The distribution of x for 10 observations. (d) The distribution of x for 25 observations. ¯ ¯ ¯

Copyright ©2013, 2010, 2007, 2004 by W. H. Freeman and Company The Basic Practice of Statistics, 6 th Edition David S. Moore, William I. Notz, Michael A. Fligner Figure 11.6 The exact distribution (dotted) and the Normal approximation from the central limit theorem (solid) for the average time needed to maintain an air conditioner, for Example The probability we want is the area to the right of 1.1.