chapter 7 sampling distributions. sampling distribution of the mean inferential statistics...

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.