Chapter 7 Sampling Distributions

Sampling Distribution of the Mean Inferential statistics –conclusions about population Distributions –if you examined every possible sample, you could put the results into a sampling distribution. “Cereal Filling” is an excellent story about inferential statistics.

Review Central Tendency Many measures. Arithmetic Mean is best, IF data or population probability distribution is normal or approximately normal. “Unbiased” –a property of statistics –if you take all possible sample means for a given sample size, the average of the sample means will equal µ.

Demo of “Unbiasedness” Table 7.1 RV = ? Finite population for demo purposes µ=? σ=?! Say that you take a sample, n = 2, with replacement. How many different x-bars are there? If you average all of them, the average = μ. This demonstrates “unbiasedness.”

Unbiased Estimator Statistics are used to estimate parameters. Some statistics are better estimators than others. We want unbiased estimators. X-bar is an unbiased estimator of µ.

Standard Error of the Mean Our estimator of µ is x-bar. X-bar changes from sample to sample, that is, x-bar varies. The variation of x-bar is described by the standard deviation of x-bar, otherwise known as the standard error of the mean.

Sampling from Normally Distributed Populations If your population is Normally distributed (ie. You are dealing with a RV that conforms to a normal probability distribution), with parameters µ and σ, and you are sampling with replacement, then the sampling distribution will be normally distributed with mean= µ and standard error = σ/  n

Central Limit Theorem Extremely important. Given large enough sample sizes, probability distribution of x-bar is normal, regardless of probability distribution of x.

7.3 Sampling Distribution of the Proportion Given a nominal random variable with two values (e.g. favor, don’t favor, etc.), code (or score) one of the values as a 1 and code the other as a 0. By adding all of the codes (or scores) and dividing by n, you can find the sample proportion.

Population Proportion The sample proportion is an unbiased estimator of the population proportion. The standard error of the proportion appears in formula 7.7, page 239. The sampling distribution of the proportion is binomial; however, it is well approximated by the normal distribution if np and n(1-p) both are at least 5. The appropriate z-score appears in formula 7.8, page 240.

Why create a frame / draw a sample? less time consuming than census less costly than census less cumbersome than census— easier, more practical

Types of Samples Figure 7.5 Nonprobability –Advantages –Disadvantages Probability (best) –Advantages –Disadvantages Simple Random Sampling

Ethical Issues Purposefully excluding particular groups or members from the “frame.” Knowingly using poor design. Leading questions. Influencing the respondent. Respondent falsifying answers. Incorrect generalization to the population.