Sampling Distribution of a Sample Proportion Lecture 27

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). n We saw that the shapes of their distributions was moving towards the shape of the normal distribution. n

Preview of the Central Limit Theorem 2 1 0 1

Preview of the Central Limit Theorem 2 1 0 1

Preview of the Central Limit Theorem 2 1 0 1

Preview of the Central Limit Theorem n 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. n Can we quantify this?

Preview of the Central Limit Theorem m= ½ 2 = 1/12 2 1 0 1

Preview of the Central Limit Theorem m= ½ 2 = 1/24 2 1 0 1

Preview of the Central Limit Theorem m= ½ 2 = 1/36 2 1 0 1

Preview of the Central Limit Theorem n 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 n 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. n Population parameters are fixed. n

Some Questions n n n 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?