Sampling Distributions Chapter 7 Central Limit Theorem Goal: Use and interpret results using the Central Limit Theorem

Goal: Use and interpret results using the Central Limit Theorem Sampling Distribution Simulation

Goal: Use and interpret results using the Central Limit Theorem Take a random sample of size n from any population with mean m and standard deviation s. When n is large, the sampling distribution of the sample mean is close to the normal distribution. How large a sample size is needed depends on the shape of the population distribution. Rule of Thumb – N=30 will guarantee normality for all shapes of population distributions

Goal: Use and interpret results using the Central Limit Theorem Uniform distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 1

Goal: Use and interpret results using the Central Limit Theorem Uniform distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 2

Goal: Use and interpret results using the Central Limit Theorem Uniform distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 3

Goal: Use and interpret results using the Central Limit Theorem Uniform distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 4

Goal: Use and interpret results using the Central Limit Theorem Uniform distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 8

Goal: Use and interpret results using the Central Limit Theorem Uniform distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 16

Goal: Use and interpret results using the Central Limit Theorem Uniform distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 32

Triangle distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 1

Triangle distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 2

Triangle distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 3

Triangle distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 4

Triangle distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 8

Triangle distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 16

Triangle distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 32

Goal: Use and interpret results using the Central Limit Theorem Inverse distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 1

Goal: Use and interpret results using the Central Limit Theorem Inverse distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 2

Goal: Use and interpret results using the Central Limit Theorem Inverse distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 3

Goal: Use and interpret results using the Central Limit Theorem Inverse distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 4

Goal: Use and interpret results using the Central Limit Theorem Inverse distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 8

Goal: Use and interpret results using the Central Limit Theorem Inverse distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 16

Goal: Use and interpret results using the Central Limit Theorem Inverse distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 32

Parabolic distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 1

Parabolic distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 2

Parabolic distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 3

Parabolic distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 4

Parabolic distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 8

Parabolic distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 16

Parabolic distribution Goal: Use and interpret results using the Central Limit Theorem Sample size 32

Goal: Use and interpret results using the Central Limit Theorem Loose ends Goal: Use and interpret results using the Central Limit Theorem An unbiased statistic falls sometimes above and sometimes below the actual mean, it shows no tendency to over or underestimate. As long as the population is much larger than the sample (rule of thumb, 10 times larger), the spread of the sampling distribution is approximately the same for any size population.

Goal: Use and interpret results using the Central Limit Theorem Loose ends Goal: Use and interpret results using the Central Limit Theorem As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean m? As the sample size increases, the mean of the observed sample gets closer and closer to m. (law of large numbers)