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Chapter 11: Bivariate Statistics Chapter 11: Bivariate Statistics and Statistical Inferenceand Statistical Inference

“Figures don’t lie, but liars figure.”

Key Concepts: Statistical Inference

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Making InferencesMaking Inferences We calculate the odds…We calculate the odds…

Crossing a busy street and not getting hit?Crossing a busy street and not getting hit? The movie will be good?The movie will be good? A child will be safe if left with parents in which A child will be safe if left with parents in which

there was previous child abuse?there was previous child abuse?

Probability (the odds, chances of)Probability (the odds, chances of) The chances of an event based on the ratio of The chances of an event based on the ratio of

favorable outcomes to total outcomes.favorable outcomes to total outcomes. Getting heads on a coin flip 1/2 = .5 (50%)Getting heads on a coin flip 1/2 = .5 (50%) Getting an Ace from cards: 4/52 = 7.7%Getting an Ace from cards: 4/52 = 7.7%

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What are the odds (con’t.)What are the odds (con’t.) Average height: male – 5’9”; female 5’5”Average height: male – 5’9”; female 5’5”

Which groups are all male and all female? Which groups are all male and all female? Is the sample representative of the population?Is the sample representative of the population?

How certain are you of your answer?How certain are you of your answer? Could all 3 groups be the same sex?Could all 3 groups be the same sex?

Group 1: 5’8, 5’6, 5’4, 5’2, 5’4, 5’5, 5’6, 5’1Group 2: 5’9, 6’1, 5’8, 5’7, 5’8, 5’9, 6’2, 5’6Group 3: 5’6, 5’8, 6’0, 5’2, 5’7, 5’5, 5’7, 5’6

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Statistical InferenceStatistical Inference

Variation – differences in behavior, Variation – differences in behavior, attitudes, values, characteristics, etc.attitudes, values, characteristics, etc. There is variation in the populationThere is variation in the population

e.g., some people like chocolate, others like vanilla.e.g., some people like chocolate, others like vanilla.

A sample is picked from the population. A sample is picked from the population. We study the sample to make inferences about the We study the sample to make inferences about the

population. population. To do so, the sample must reflect the population in To do so, the sample must reflect the population in

the characteristics under study.the characteristics under study. If 30% prefer chocolate in the sample, we’d like to If 30% prefer chocolate in the sample, we’d like to

conclude that 30% prefer chocolate in the conclude that 30% prefer chocolate in the population.population.

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Statistical Inference (con’t.)Statistical Inference (con’t.) BUT, sometimes the sample does not reflect the BUT, sometimes the sample does not reflect the

population – the extent to which it doesn’t is population – the extent to which it doesn’t is SAMPLE ERROR.SAMPLE ERROR. We can calculate the chances that the relationship We can calculate the chances that the relationship

between variables in the sample is due to sample between variables in the sample is due to sample error. error.

Influences on sample errorInfluences on sample error LuckLuck – someone wins the lottery even if the odds are – someone wins the lottery even if the odds are

40 million to 1.40 million to 1. Sample sizeSample size – smaller samples will have more – smaller samples will have more

chances of error.chances of error. HomogeneityHomogeneity – less variation in the population yields – less variation in the population yields

smaller sample error.smaller sample error.

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Hypothesis TestingHypothesis Testing Testing the relationship between two or more Testing the relationship between two or more

variables.variables. Statistical tests are used to find the probability Statistical tests are used to find the probability

that the relationship between variables is due that the relationship between variables is due to sampling error or to chance.to sampling error or to chance.

Type Example

Null hypothesis (Ho) – No relationship There is no relationship between income and mental health.

Two-tailed hypothesis (H1) – There is

a relationship

There is a relationship between income and mental health.

One-tailed hypothesis (H1) –

Directional relationship

The greater the income the greater the mental health.

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Statistical Inference (con.)Statistical Inference (con.) pp-value-value

The probability that a relationship between The probability that a relationship between variables or a mean difference found in a sample variables or a mean difference found in a sample is a result of sample error.is a result of sample error.

pp =.05 means there is a 5% chance that the =.05 means there is a 5% chance that the relationship found in the sample is a result of relationship found in the sample is a result of sample error.sample error.

pp =.05 means there is a 95% that the relationship =.05 means there is a 95% that the relationship is NOT due to sample error, and actually reflects is NOT due to sample error, and actually reflects the differences in the population.the differences in the population.

Rejection levelRejection level: If the : If the pp value is <.05, we reject value is <.05, we reject the null hypothesis and accept the alternative the null hypothesis and accept the alternative hypothesis. (Why .05? – Convention).hypothesis. (Why .05? – Convention).