Sampling Distribution of a Sample Proportion Lecture 27 Sections 8.1 – 8.2 Tue, Mar 6, 2007

Preview of the Central Limit Theorem We looked at the distribution of the average of 1, 2, and 3 uniform random variables U(0, 1). We saw that the shapes of their distributions was moving towards the shape of the normal distribution.

Preview of the Central Limit Theorem 2 1 1

Preview of the Central Limit Theorem 2 1 1

Preview of the Central Limit Theorem 2 1 1

Preview of the Central Limit Theorem Some observations: Each distribution is centered at the same place, ½. The distributions are being “drawn in” towards the center. That means that their standard deviation is decreasing. Can we quantify this?

Preview of the Central Limit Theorem 2 = ½ 2 = 1/12 1 1

Preview of the Central Limit Theorem 2 = ½ 2 = 1/24 1 1

Preview of the Central Limit Theorem 2 = ½ 2 = 1/36 1 1

Preview of the Central Limit Theorem This tells us that a mean based on three observations is much more likely to be close to the population mean than is a mean based on only one or two observations.

Parameters and Statistics THE PURPOSE OF A STATISTIC IS TO ESTIMATE A POPULATION PARAMETER. A sample mean is used to estimate the population mean. A sample proportion is used to estimate the population proportion.

Parameters and Statistics Sample statistics are variable. Population parameters are fixed.

Some Questions We hope that the sample proportion is close to the population proportion. How close can we expect it to be? Would it be worth it to collect a larger sample? If the sample were larger, would we expect the sample proportion to be closer to the population proportion? How much closer?