Chapter 5 Section 2 Part 1

What samples tell us? Gallup Poll (lottery Participation) A new Gallup Poll Social Audit on gambling shows that 57% of Americans have bought a lottery ticket in the last 12 months, making lotteries by far the favorite choice of gamblers. Reading farther, we find that they talked with 1523 randomly selected adults to reach these conclusions. There are 200 million people in the world.

How can 1523 people, even randomly selected, tell us about the habits of 200 million people? Does this mean that 57% of ALL Americans have bought lottery tickets?

Vocabulary A Parameter a number that describes the population. A parameter is a fixed number, but in practice we do not know its value. Typically the mean. A Statistic is a number that describes a sample. The value of the statistic is known when we have taken a sample, but it can change from sample to sample. We often use a statistic to estimate an unknown parameter. A “parameter” is to a “population” as a “statistic” is to a “sample”

Example: A recent Gallup Poll interviewed a random sample of 1523 adults. Of these, 868 bought a lottery ticket in the past year. Population: Parameter: Sample: Statistic: Adult Americans Proportion who bought lottery tickets 1523 adults randomly chosen .57 or 57%

Statistics The proportion of a sample is the statistic – we abbreviate it as 𝑝 (read as “p-hat”)

Example A random sample of 1000 people who signed a card saying they intended to quit smoking on Nov. 20, 1995, and were contacted in June 1996. It turned out that 210 of the sampled individuals has not smoked over the past six months. Specify the population of interested, parameter of interest, the sample, and the sample statistic in this problem.

Sampling Variability Does random sampling guarantee “good” results?

Results First, when we take random samples it eliminates bias. Second, if we take lots of random samples of the same size from the sample population, the variation from sample to sample would follow predictable pattern. This predictable pattern shows the results of bigger samples are less variable than the results of smaller samples.

Two Types of Errors in Estimation BIAS is consistent, repeated deviation of the sample statistic from the population parameter in the same direction when we take many samples. Variability describes how spread out the values of the sample statistic are when we take many samples. Large variability means that the results of sampling is not respectable.

A good sampling method has both small bias and small variability

Example Bulls eye

Managing Bias and Variability To reduce bias, use random sampling. When we start with a list of the entire population, simple random sampling produces unbiased estimates – the values of a statist computer from a SRS neither consistently overestimate nor consistently underestimate the value of the population parameter.

Managing Bias and Variability To reduce the variability of an SRS, use a larger sample. You can make the variability as small as you want by taking a large enough sample.

Homework