Foundations of Inferential Statistics PADM 582 University of La Verne Soomi Lee, Ph.D

Overview Review Descriptive Statistics Some in-class exercise 1.Hypothesis 2.Probability 3.Significance

Why should we care? Quantitative (interval, ratio) Ordinal Nominal (qualitative) Levels of Measurement Respondent ID Months in Unemployment Level of Satisfaction in New Job Training Program (scale 1-5) Gender (1=female; 0=male)

Why should we care? 1.List the Sample 2.Frequency Table 3.Summary Statistics – Central tendency: mean, median, mode – Dispersion (spread): standard deviation, range, variance – Skewness, kurtosis 4.Graphs and Charts – Histogram (quantitative) – Bar, column, pie charts (ordinal, nominal) – Line chart: historical data (time series data) Correlation: description of the relationship between two variables – Correlation coefficient, Correlation of Determination – Visualization: scatter plot Ways to Summarize Data

Why should we care? Should the department send the dean the mean age or the median age? In-class Exercise 1 MemberAge Williamson64 Campbell31 Gonzales65 Marquez27 Seymour35 Sandoval40 Weber33

Why should we care? Who is Correct? In-Class Exercise 2 Case WorkerCase Load Williamson43 Campbell57 Gonzales35 Marquez87 Seymour36 Sandoval93 Weber45 Kim48 Meier41 Becker40

Why should we care? What does this number mean to the chief of custodial engineer? Skewness = In-Class Exercise 3

Why should we care? PoliceFire Mean=603Mean=601 In-Class Exercise 4 There is really no fairness issue to worry about. Is it true?

Summary Statistics

What you will learn in Chapter 7 1.The difference between samples and populations (again) 2.The importance of… The null hypothesis The research hypotheses 3.How to judge a good hypothesis Hypothesis (Ch.7)

What is a hypothesis? An “educated guess” Their role is to reflect the general problem statement or question that is driving the research Translates the problem or research question into a form that can be tested. What is Hypothesis?

Samples and Populations Population – The large group to which you would like to generalize your findings Sample – The smaller, representative group of the population that is used to do the research Sampling error – a measure of how well a sample represents the population Samples and Populations

The Null Hypothesis Statements that contain two or more things that are unrelated to one another H 0 :  1 =  2 – The starting point and is accepted as true without knowing more information – Benchmark to compare actual outcomes The Null Hypothesis

The Research Hypothesis

Null HypothesisResearch Hypothesis No relationship between variablesRelationship between variables Refers to the populationRefers to the sample Indirectly testedDirectly tested Written using Greek symbolsWritten using Roman symbols Implied hypothesisExplicit hypothesis Differences between Null and Research Hypotheses

What Makes a Good Hypothesis? Stated in a declarative form rather than a question Defines an expected relationship between variables Reflects the theory or literature on which they are based Brief and to the point Testable – includes variables that can be measured What Makes a Good Hypothesis?

Stated in a declarative form rather than a question Defines an expected relationship between variables Reflects the theory or literature on which they are based Brief and to the point Testable – includes variables that can be measured What Makes a Good Hypothesis?

Group Work

What you will learn in Chapter 8 1.Understanding probability is basic to understanding statistics 2.Characteristics of the “normal” curve – i.e. the bell-shaped curve 3.All about z scores – Computing them – Interpreting them Probability

Why Probability? Basis for the normal curve – Provides basis for understanding probability of a possible outcome Basis for determining the degree of confidence that an outcome is “true” – Example: Are changes in student scores due to a particular intervention that took place or by chance alone? Why Probability?

Visual representation of a distribution of scores Three characteristics… 1.Mean, median, and mode are equal to one another 2.Perfectly symmetrical about the mean 3.Tails are asymptotic (get closer to horizontal axis but never touch) The Normal Curve (the Bell-Shaped Curve)

The Normal Curve The Normal Curve (the Bell-Shaped Curve)

In general, many events occur right in the middle of a distribution with few on each end. The Normal Curve (the Bell-Shaped Curve)

More Normal Curve 101

For all normal distributions… – Almost 100% of scores will fit between -3 and +3 standard deviations from the mean. – So…distributions can be compared – Between different points on the X-axis, a certain percentage of cases will occur. More Normal Curve 101

What’s Under the Curve? What’s under the Curve?

The z Score A standard score that is the result of dividing the amount that a raw score differs from the mean of the distribution by the standard deviation. What about those symbols? The z Score

A common statistical way of standardizing data on one scale so a comparison can take place is using a z-score. The z-score is like a common yard stick for all types of data. The z Score

Using the Computer Calculating z Scores

The z Score Scores below the mean are negative (left of the mean) and those above are positive (right of the mean) A z score is the number of standard deviations from the mean z scores across different distributions are comparable The z Score

What z Scores Represent The areas of the curve that are covered by different z scores also represent the probability of a certain score occurring. So try this one… – In a distribution with a mean of 50 and a standard deviation of 10, what is the probability that one score will be 70 or above? What z Score Represent

What z Scores Really Represent Knowing the probability that a z score will occur can help you determine how extreme a z score you can expect before determining that a factor other than chance produced the outcome Keep in mind… z scores are typically reserved for populations What z Score Really Represent

Hypothesis Testing & z Scores Any event can have a probability associated with it. – Probability values help determine how “unlikely” the event might be – The key - less than 5% chance of occurring and you have a significant result Hypothesis Testing and z Scores

Using the Computer Group Work

What you will learn in Chapter 9 1.What significance is and why it is important – Significance vs. Meaningfulness 2.Type I Error 3.Type II Error 4.How inferential statistics works Statistical Significance

The Concept of Significance Any difference between groups that is due to a systematic influence rather than chance – Must assume that all other factors that might contribute to differences are controlled The Concept of Significance

If Only We Were Perfect… Significance level – The risk associated with not being 100% positive that what occurred in the experiment is a result of what you did or what is being tested The goal is to eliminate competing reasons for differences as much as possible. Statistical Significance – The degree of risk you are willing to take that you will reject a null hypothesis when it is actually true. If Only We are Perfect…

The World’s Most Important Table Different Types of Errors

Type I Errors (Level of Significance) The probability of rejecting a null hypothesis when it is true Conventional levels are set between.01 and.05 Usually represented in a report as p <.05 Type I Errors

Type II Errors The probability of accepting a null hypothesis when it is false As your sample characteristics become closer to the population, the probability that you will accept a false null hypothesis decreases Type II Errors

Significance Versus Meaningfulness A study can be statistically significant but not very meaningful Statistical significance can only be interpreted for the context in which it occurred Statistical significance should not be the only goal of scientific research – Significance is influenced by sample size…we’ll talk more about this later. Significance vs. Meaningfulness

How Inference Works A representative sample of the population is chosen. A test is given, means are computed and compared A conclusion is reached as to whether the scores are statistically significant Based on the results of the sample, an inference is made about the population. How Inference Works

Test of Significance 1. A statement of the null hypothesis. 2. Set the level of risk associated with the null hypothesis. 3. Select the appropriate test statistic. 4. Compute the test statistic (obtained) value 5. Determine the value needed to reject the null hypothesis using the appropriate table of critical values 6. Compare the obtained value to the critical value 7. If obtained value is more extreme, reject the null hypothesis 8. If obtained value is not more extreme, accept the null hypothesis Test of Significance

Next Week Homework 3 due: next week in class Extra credit homework (eligible only for those who got scores below 50 on hw1) due also next week in class