Examine characteristics of a sample and make inferences about the population

Transcription

1 Chapter 11 Introduction to Inferential Analysis Learning Objectives Understand inferential statistics Explain the difference between a population and a sample Explain the difference between parameter and a statistic Understand the expected value and standard error Learning Objectives Calculate and interpret a Z-score Understand the normal curve and the central limit theorem Examine probability Determine confidence intervals Inferential analysis Examine characteristics of a sample and make inferences about the population Collecting data on the entire population is too costly, too time consuming, and impossible Make decision with probability of error about whether sample characteristic is different from population characteristic Terminology and Assumptions Terminology: Population: entire group of study Parameter: a characteristic of the population Sample: subset of population drawn to allow statistical analysis Statistic: a characteristic of the sample

2 Terminology and Assumptions Terminology: Inference: performing analysis on sample and concluding that the findings apply to the population Estimate: Result of sample analysis of the population parameters Expected Value: estimated population parameters Terminology and Assumptions Terminology: Standard Error: measure of the variation of a statistic around the parameter it is estimating Terminology and Assumptions Assumptions: Sample is from same population as inferences Population and sample normally distributed Random sample of data was taken Probability Can estimate probability of: Event occurring Sample statistic matching population parameter Number of times an event can occur vs. number of times any event can occur Important for inferential analysis because it represents probability of making a wrong decision about null hypothesis Probability Range: 0 to 1

3 0 = impossible 1 = imminent Two rules: Addition rule: probability of one event or another event Multiplication rule: probability of one event and another event Sampling Three important populations: Parent population One or more samples are drawn from this Target population Population to which findings generalized Study population Sample actually drawn Sampling Sampling units represent data Cases will be examined Sampling frame Actual list of sampling units Sampling unit All elements chosen in some stage of sampling Sampling Types of sampling units Primary sampling units

4 Secondary sampling units Final sampling units Element Characteristic about which data is collected Provides basis for analysis Sampling Probability Each element has an equal chance of being drawn Non-Probability Based on study characteristics or convenience for research Probability Sampling Preferred method for inferential analysis Difficult to draw except small populations Each element must have the same probability of being drawn, and probability cannot be 0 Simple Random Sample Ideal sample drawing method Elements have equal and known chance of selection Simple Random Sample Procedures: Need complete list of all elements in population Number consecutively ( 1 to N ) Computer generates table Sample taken randomly

5 Need sampling with replacement Systematic Sampling Can be same as simple random sample if elements have equal and known chance of selection Establish sampling frame as with simple random sample Choose elements in systematic fashion Every 3 rd, 5 th, 10 th, based on sample size desired Must not stop sampling before reaching end of sampling frame Stratified Sampling Purpose Uses known sample information to ensure adequate data gathered on all elements How Divide population into groups Based on variables to stratify Random or non-random characteristics sample Proportionate Disproportionate Cluster or Multistage Sampling Cluster sampling Divide sampling frame into clusters Often subdivided Multistage sampling If clustered or stratified Secondary sampling chosen

6 Both create levels of sampling units Nonprobability Sampling More convenient Less expensive Easier to collect Problems Probability not known Randomization cannot control inaccuracies Purposive Sampling Chosen based on researcher s knowledge of population Belief is that sample is representative of population Quota Sampling Non-probability stratified sample Divide population into groups based on desired characteristics Selection based on stratification Seeks resemblance with proportions of desired characteristics in population Oversampling: more elements are chosen than their proportion in the population Snowball Sampling For difficult subjects No population inferences Provide detail on chosen sample Accidental or Convenience Sampling

7 Accidental Whoever happens by Convenience Using people nearby Easy to do but difficult to make inferences Normal Curve Typically disconnect between sample and population Works through principle of sampling distributions Sample should be representative of population Normal Curve Sampling Distributions Probability distributions specifically for inferential analysis Expected characteristics of large number of samples Makes use of central limit theorem Allows use of single sample Sample considered normal Uses principles of normal curve: large samples Sampling Distributions Sampling distribution of the means Means often estimate population characteristics Represents measure of central tendency (expected value)

8 Standard error Variation of sample around population parameter Central Limit Theorem Allows estimates and generalizations Based on inferred population Normal distribution achieved by increasing sample size Based on sampling distribution of the means and normal curve Sample mean approximates population mean Better estimate of population mean Confidence Intervals Point estimation Finding expected value based on sample data Interval estimation Using confidence intervals to address sampling error Establishes range of values toward true population Calculating Confidence Intervals Interpreting Confidence Intervals Range of interval is function of confidence level desired and sample size Higher confidence needs wider interval Conclusion Concepts of normal curve, probability, and sampling distributions can be applied to hypothesis testing

9 Sampling distributions give researchers confidence that the expected value will approximate the population parameter Normal curve facilitates use of sampling distributions through probabilities Confidence intervals expand ability to estimate populate parameter