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12/14/2018

A Brief Introduction to Probability

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Probability − Likelihood of occurrence of outcome of an

event given all possible outcomes

A Brief Introduction to Probability

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Probability − What’s the likelihood of randomly

selecting the fastest checkout line at a grocery store with 5 checkout lanes?

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A Brief Introduction to Probability

• The sum of the probabilities of all possible outcomes is 1.00 (100%)

• Addition rule: – When we have mutually exclusive outcomes, the probabilities of the outcomes can be summed to find a combined probability

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Why is Probability Important to Researchers?

• Because research is typically conducted with samples rather than populations, researchers rely on probability to evaluate data collected from samples

• Sampling error – The difference between statistics calculated from a sample and those from the population

– Samples are imperfect representations of the population

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Applying Probability to Normal Distributions

• Percentages of scores in the normal curve can be re‐expressed as probabilities.

• For example, – The "percentage of scores between z = .00 and z = 2.24” is the same as the “probability that a randomly selected z‐ score will be between z = .00 and z = 2.24”

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Applying Probability to Binomial Distributions

• Binomial distributions are based on variables with only two categories

– Gender (male, female)

– Test response (correct, incorrect)

– Coin (heads, tails)

• Probability is labeled as p and q p = p(heads) and q = p(tails)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

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Applying Probability to Binomial Distributions

• The probabilities of outcomes in a binomial distribution become increasingly normal in shape with larger sample sizes

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to Hypothesis Testing

• Research hypotheses vs. statistical hypotheses

– Research hypotheses: focus on verbal expression of concepts

– Statistical hypotheses: focus on numerical expressions of relationships between variables

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Introduction to Hypothesis Testing

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to Hypothesis Testing: Null and Alternative Hypothesis

• A set of hypotheses that present two mutually exclusive conclusions about data

• Null hypothesis (H0) – Proposes that the hypothesized change, difference, or relationship does not exist

– Football coin toss example with a total of 12 games:

H0:  = 6

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Introduction to Hypothesis Testing: Null and Alternative Hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Alternative hypothesis (H1) − Proposes that the hypothesized change,

difference, or relationship does exist. − Football coin toss example:

H1 : μ ≠ 6

Introduction to Hypothesis Testing

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Introduction to Hypothesis Testing: Making a Decision

• What would lead us to reject the null hypothesis (H0)?

– Can’t simply compare the statistic calculated from a sample with the hypothesized population parameter

• Recall the discussion of sampling error

– We look for evidence that has a low probability of occurring when H0 is true

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to Hypothesis Testing: Making a Decision

• Defining a “low” probability – Alpha (): probability of a statistic used to make the decision whether to reject H0

– Conventionally,  = .05, or a 5% chance of the statistic occurring when H0 is true

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Introduction to Hypothesis Testing: Making a Decision

• Identifying values of a statistic with low probability – What values have a combined probability < ? – Reading a table

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to Hypothesis Testing: Making a Decision

• Region of rejection – Values of a statistic whose combined probability is low enough to lead to the decision to reject H0

• Region of non‐rejection – Values of a statistic whose combined probability is not low enough to lead to the decision to reject H0

• Critical value – Value of a statistic that separates the two regions

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Introduction to Hypothesis Testing: Making a Decision

• For  = .05 and N = 12 games, critical values = 3 wins and 9 wins

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to Hypothesis Testing: Making a Decision

• Stating a decision rule – If the value of the statistic lies beyond the critical values (is in the region of rejection), reject the null hypothesis; otherwise, do not reject the null hypothesis.

– Example: If the number of wins in 12 Super Bowls is fewer than 3 or greater than 9, reject H0; otherwise do not reject H0.

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Introduction to Hypothesis Testing: Making a Decision

• Calculating a value of the statistic – Calculate a value of a statistic that will be compared with the critical values

– (Note: the statistic that’s calculated will change throughout the course, depending on the research situation)

• Make a decision whether to reject H0 – Compare the statistic to the critical value

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to Hypothesis Testing

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Introduction to Hypothesis Testing: Drawing a Conclusion

• The conclusion should state – The sample from whom the data were collected

– The value of the statistic calculated from the data

– The decision about the null hypothesis

– The probability of the statistic

Introduction to Hypothesis Testing

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

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Introduction to Hypothesis Testing: Relating to the Research Hypothesis

• Does the statistical analysis support the original research hypothesis?

Issues Related to Hypothesis Testing: “Proof”

• Because hypothesis testing is based on probability, it cannot provide “proof”

– Proof implies absolute certainty

– Hypothesis testing can only provide support or lack of support for a hypothesis

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Issues Related to Hypothesis Testing: Errors

• Understand that it’s possible that the decision made about the null hypothesis could be incorrect

– Rejecting the null hypothesis when we shouldn’t

– Not rejecting the null hypothesis when we should

• These two errors will be discussed later in the book

Issues Related to Hypothesis Testing: Factors that Affect the Decision

• Sample size – The larger the sample size, the greater the likelihood of rejecting H0

• Alpha – The smaller the value of , the lower the likelihood of rejecting H0

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Issues Related to Hypothesis Testing: Factors that Affect the Decision

• Directionality of the alternative hypothesis. − Non‐directional hypotheses do not

specify the direction of change, difference, relationship

− Directional hypotheses specify the direction of change, difference, relationship

Issues Related to Hypothesis Testing: Factors that Affect the Decision

Non‐directional hypotheses: Regions of rejection in both ends of the distribution (two‐tailed)

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Issues Related to Hypothesis Testing: Factors that Affect the Decision

Directional hypotheses: Region of rejection in only one end of the distribution (one‐tailed)

Looking Ahead

• Now that you are familiar with the steps of hypothesis testing, we will begin to talk about different statistical procedures that compare sample means with population means