Chapter 9 Sampling Distributions This chapter prepares us for the study of Statistical Inference by looking at the probability distributions of sample proportions and sample means.

Population→parameter Sample→statistic Parameters are fixed. Statistics vary.

Statistic & Parameter The government's Current Population Survey contacted a sample of 113,146 households in March Their mean income was $60,528. Describe the statistic and parameter of interest, µ and x bar.

Statistic & Parameter The Gallup Poll asked a random sample of 515 US adults whether they believe in ghosts. Of the respondents, 160 said “Yes”. Identify the statistic and parameter, p and phat.

Sampling Variability: the value of a statistic varies in repeated random sampling

Simulating Sampling Variability Thousands of travelers pass through Guadalajara airport each day. Before leaving the airport, each passenger must pass through the Customs inspection area and press a button. If the light shows green, the person can go ahead. If the light shows red, the persons bags are searched. Set up a simulation that represents a sample of size 100 from the population of travelers if the Customs officers claim the probability of a green light is 0.70.

1000 SRS's of size 100 drawn from a population with p = 0.7 An approximation of the sampling distribution of phat.

An exact Sampling Distribution Construct a sampling distribution for taking an SRS of size 2 from the random number table B. remember...

Television executives and companies who advertise onTV are interested in how many viewers watch particular TV shows. Suppose that the true proportion of U.S. Adults who watched Survivor: Guatemala is p=0.37. The following shows the results of drawing 1000 SRS's of size n=100 from this population. Sampling Distribution of phat

Describe the distribution of the proportions of samples who watched Survivor: Guatemala from sample sizes of 100. Sampling Distribution of phat

Is a sample of 100 people produce a trustworthy estimate of the population proportion? Sampling Distribution of phat

The Gallup Poll actually took a sample of 1000 people and asked them if they watched Survivor. Sampling Distribution of phat Describe the distribution of the proportions of samples who watched Survivor: Guatemala from sample sizes of 1000.

Here is the distribution drawn on an expanded scales so that it better shows the shape. Sampling Distribution of phat

How trustworthy is the sample proportion phat as an estimator of the population proportion p in each case? In fact... The mean of the sampling distribution (think of all possible samples, not just 1000 samples) is exactly equal to the parameter in the population.

Were the Survivor statistics unbiased? Bias concerns the center of the sampling distribution. Unbiased – the statistic has no systematic tendency to overestimate or underestimate the parameter. The sample statistic phat is an unbiased estimator for the population parameter p. The sample statistic xbar is an unbiased estimator for the population parameter μ.

The spread of the sampling distribution depends on the sample size, not the size of the population!

Page 575 last paragraph Good news for national samples, bad news for the local samplers of San Francisco

Properly chosen statistics computed from random samples of sufficient size will have low bias and low variability