This document provides an overview of inferential statistics, including key terminology like population, sample, parameter, statistic, and estimate. It discusses the central limit theorem and how the sampling distribution of means becomes normal for large sample sizes. The document covers point estimation and interval estimation using confidence intervals. It explains how to construct confidence intervals by adding and subtracting values like the standard error of the mean from the sample statistic. The level of confidence, like 95% or 99%, determines how wide the interval needs to be to capture the population parameter.

1. Topics: Inferential Statistics
• Inference
• Terminology
• Central Limit Theorem
• Estimation
– Point Estimation
– Confidence Intervals
• Hypothesis Testing

2. 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

3. How Statistical Inference Works

4. 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.)

5. 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.)

6. Basic Statistical Symbols

7. Basic Terminology (con’t)
• Estimate: a number computed by using the
data collected from a sample
• Estimator: formula used to compute an
estimate

8. The Process of Estimation

9. Types of Samples
• Probability
– Simple Random Samples
– Simple Stratified Samples
– Systematic Samples
– Cluster Samples
• Non Probability
– Purposive Samples
– Convenience Samples
– Quota Samples
– Snowball Samples

10. Limits on Inferences and Warnings
• Response Rates
• Source of data
• Sample size and sample quality
• “Random”

11. Estimation
• Point Estimation
• Interval estimation
– Sampling Error
– Sampling Distribution
– Confidence Intervals

12. 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

13. 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

14. 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

15. Sampling Distribution of Mean

16. Sampling Distribution of Means and
Standard Error of the Means
u
mu
+2sem
-2sem +1sem
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Population mean

17. 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

18. Sampling Distribution of 1000 Sample
Means
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1.5 pts
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minus
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4.5 pts

19. 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

20. Factors Affecting Confidence Intervals

21. 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

22. 95%Confidence Interval
u
mu
+1.96sem
-1.96sem
95%
-2.58sem
+2.58sem
95 times out of 100 the interval constructed
around the sample mean will capture
the population mean. 5 times out of 100 the
interval will not capture the population mean

23. 99%Confidence Interval
u
mu
99%
-2.58sem
+2.58sem
99 times out of 100 the interval constructed
around the sample mean will capture
the population mean. 1 time out of 100 the
interval will not capture the population mean

24. Effects of Sample Size

25. 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