Topics: Inferential Statistics

• Inference

• Terminology

• Central Limit Theorem

• Estimation– Point Estimation– Confidence Intervals

• Hypothesis Testing

Inferential Statistics

• Research is about trying to make valid inferences

• Inferential statistics: the part of statistics that allows researchers to generalize their findings beyond data collected.

• Statistical inference: a procedure for making inferences or generalizations about a larger population from a sample of that population

How Statistical Inference Works

Basic Terminology

• Population: any collection of entities that have at least one characteristic in common

• Parameter: the numbers that describe characteristics of scores in the population (mean, variance, s.d., etc.)

Basic Terminology (cont’d)

• Sample: a part of the population

• Statistic: the numbers that describe characteristics of scores in the sample (mean, variance, s.d., correlation coefficient, reliability coefficient, etc.)

Basic Statistical Symbols

Basic Terminology (con’t)

• Estimate: a number computed by using the data collected from a sample

• Estimator: formula used to compute an estimate

The Process of Estimation

Types of Samples

• Probability– Simple Random Samples

– Simple Stratified Samples

– Systematic Samples

– Cluster Samples

• Non Probability– Purposive Samples

– Convenience Samples

– Quota Samples

– Snowball Samples

Limits on Inferences and Warnings

• Response Rates

• Source of data

• Sample size and sample quality

• “Random”

Estimation

• Point Estimation

• Interval estimation– Sampling Error– Sampling Distribution– Confidence Intervals

Interval Estimation

• Interval Estimation: an inferential statistical procedure used to estimate population parameters from sample data through the building of confidence intervals

• Confidence Intervals: a range of values computed from sample data that has a known probability of capturing some population parameter of interest

Sampling Error

• Samples rarely mirror exactly the population

• The sample statistics will almost always contain sampling error

• The magnitude of the difference of the sampling statistic from the population parameter

Sampling Distribution

• Sampling Distribution: a theoretical distribution that shows the frequency of occurrence of values of some statistic computed for all possible samples of size N drawn from some population.

• Sampling Distribution of the Mean: A theoretical distribution of the frequency of occurrence of values of the mean computed for all possible samples of size N from a population

Sampling Distribution of Mean

Sampling Distribution of Means and Standard Error of the Means

umu

+2sem-2sem +1sem-1sem-3sem +3sem

Population mean

Central Limit Theorem

• The sampling distribution of means, for samples of 30 or more:– Is normally distributed (regardless of the shape of the

population from which the samples were drawn)

– Has a mean equal to the population mean, “mu” regardless of the shape population or of the size of the sample

– Has a standard deviation--the standard error of the mean--equal to the population standard deviation divided by the square root of the sample size

Sampling Distribution of 1000 Sample Means

Ave. IQ of 5000 4th graders also Ave. of 1000 sample averages

Ave. plus 3.0 pts

Ave minus 3.0 pts

Ave. plus 1.5 pts

Ave minus 1.5 pts

Ave minus 4.5 pts

Ave. plus 4.5 pts

Confidence Intervals

• A defined interval of values that includes the statistic of interest, by adding and subtracting a specific amount from the computed statistic

• A CI is the probability that the interval computed from the sample data includes the population parameter of interest

Factors Affecting Confidence Intervals

Various Levels of Confidence

• When population standard deviation is known use Z table values:– For 95%CI: mean +/- 1.96 s.e. of mean– For 99% CI: mean +/- 2.58 s.e. of mean

• When population standard deviation is not known use “Critical Value of t” table– For 95%CI: mean +/- 2.04 s.e. of mean– For 99% CI: mean +/- 2.75 s.e. of mean

95%Confidence Interval

umu

+1.96sem-1.96sem

95%

-2.58sem +2.58sem

95 times out of 100 the interval constructedaround the sample mean will capturethe population mean. 5 times out of 100 the interval will not capture the population mean

99%Confidence Interval

umu

99%

-2.58sem +2.58sem

99 times out of 100 the interval constructedaround the sample mean will capturethe population mean. 1 time out of 100 the interval will not capture the population mean

Effects of Sample Size

Process for Constructing Confidence Intervals

• Compute the sample statistic (e.g. a mean)• Compute the standard error of the mean• Make a decision about level of confidence that is

desired (usually 95% or 99%)• Find tabled value for 95% or 99% confidence

interval• Multiply standard error of the mean by the tabled

value• Form interval by adding and subtracting calculated

value to and from the mean