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• The lecture covers samples and populations, statistical significance, one-tailed and two-tailed tests, and Type I and II errors.

Recap

• A sample's correlation is represented by Pearson's r.
• The range of possible correlation values is between -1 and +1.
• A positive correlation (+) means that scores on one variable increase/decrease as scores on another variable do as well.
• A negative correlation (-) means that as scores on one variable increase, scores on another variable decrease, and vice versa.
• Weak correlations range from r = 0 to ± 0.29.
• Moderate correlations range from r = ± 0.3 to ± 0.59.
• Strong correlations range from r = ± 0.6 to ± 1.00.
• Example: r = -0.4 is stronger than r = +0.3.

Inferential Statistics

• In statistics, a sample of scores make general statements or draw conclusions beyond that sample.
• This is called inferential statistics and involves drawing inferencesabout all scores in the population from just a sample of those scores.

Samples and Populations

• A sample is a small number of scores selected from the entirety of scores.
• A population is the entire set of scores.
• A sample is a small set, or a subset, taken from the full set or population of scores.
• Population and sample both refer to scores on a variable.
• It is often not feasible to measure the population of scores because it is infinite.

Random Samples

• Statistical inference assumes samples are drawn at random from the population.
• Obtaining a random sample of scores entails selecting scores so that each score in the population has an equal chance of being selected.
• Random samples can be selected using manual random number tables, electronic random number generators, or similar.
• Random samples typically have a mean close to the population mean; a random sample mean that is very different from the population mean is relatively rare.

Statistical Significance

• Psychologists are interested in sample means unlikely to occur through random sampling.
• The extreme 5% of the sample are of interest as being significant.
• Significance in statistics means that the sample means are very different from the population from which it was drawn.
• Significance at the 5% level indicates the sample score lies within the 5% of samples that are most different from the population.
• The 5% is obtained from looking at the extreme lower 2.5% and the extreme upper 2.5%.
• Scores in the middle 95% are likely.
• A score from a sample with only a 1 in 20 chance of occurring is unlikely to represent the population.

Null Hypothesis

• The null hypothesis (H0) always states no difference/relationship between population values (e.g. means) or between two variables.
• H0 = There is no relationship between the two variables being measured.

Statistical Significance and the Null Hypothesis

• The null hypothesis defines a population having no relationship between two variables (i.e., the middle 95%).
• A decision is made whether it is possible that the sample comes from this population, defined by the null hypothesis.
• If the sample is unlikely to come from the middle 95%, the possibility that the null hypothesis is true is rejected.

P-Values

• A p-value finds the probability of a result occurring if the null hypothesis is true.
• The null hypothesis states there is no relationship between the two variables being studied (one variable does not affect the other); the results derive due to chance and are not statistically significant.
• The alternative hypothesis states that the independent variable did affect the dependent variable and the results are significant in terms of supporting the theory being investigated (i.e., not due to chance).
• A p-value, or probability value, describes how likely it is your data would have occurred by random chance (i.e., the null hypothesis is true).
• Statistical significance is expressed as a p-value between 0 and 1; the smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
• A p-value less than 0.05 (p < .05) is statistically significant andindicates strong evidence against the null hypothesis, since there is less than a 5% probability that the null is correct (and the results are random).
• A p-value higher than 0.05 (p > 0.05) is not statistically significant and indicates strong evidence for the null hypothesis; in this case you fail to reject the null hypothesis.
• A statistically significant result cannot prove a research hypothesis correct, because that implies 100% certainty.
• Results are said to "provide support for" or "give evidence for" a research hypothesis, due to the slight probability that the results occurred by chance and the null hypothesis was correct (less than 5%).

Statistical Significance and Correlation

• A statistically significant relationship is unlikely to have occurred in the sample if there's no relationship in the population.
• Whether a result is unlikely to happen by chance is important in establishing cause-and-effect relationships.
• A correlation can be weak but still statistically significant, meaning the association is small, but nonzero.

Reporting P-Values in APA Style

• A p-value is reported with the exact value to three decimal places (e.g., p = .031).
• Report p values less than .001 as p < .001.
• Use italics for p since p is always italicized.
• The opposite of statistically significant is "not statistically significant", not "insignificant".

Type I and Type II Errors

• Type I Error: Deciding the null hypothesis is false when actually true (a false positive).
• Type II Error: Deciding the null hypothesis is true when actually false (a false negative).
• In psychology a narrow threshold for statistical significance(p < .05) is used to reduce the likelihood that Type I or Type II errors are made.

One-Tailed and Two-Tailed Significance Testing

• A one-tailed test specifies the direction of the hypothesis, e.g. 'Those who attend 80% of PS219 lectures throughout term will perform better than those who do not.''
• Two-tailed tests allow the hypothesis to go in either direction e.g. 'There will be a difference in end of term grades between those who attend 80% of PS219 lectures throughout term than those who do not.'
• Two-tailed tests are more widely used.

Problems with P-Values

• Problem #1: P Hacking - Inappropriate manipulation of data to produce a statistically significant result.
  • Making up data points
  • Removing data points
  • Altering existing data points
  • Running lots of statistical tests until you find one that produces a statistically significant result.
• Problem #2: p-values don't show absolute certainty
  • A statistically significant result cannot prove that a research hypothesis is correct (as this implies 100% certainty).
  • P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone."
• Problem #3: p < .05 does not imply a strong effect
  • A p-value describes how likely it is your data would have occurred by random chance (i.e. thatthe null hypothesis is true), butdoes not measure how big the association or the difference is. Solution: Investigate effect sizes to look at the strength of a difference or association.