Data Analysis (continued)

Analyzing the Results of Research Investigations Two basic ways of describing the results Two basic ways of describing the results 1. Descriptive statistics (n, %, mean, sd) 2. Inferential statistics 1. Correlations/regressions 2. Comparing group means (t-tests: t, ANOVA: F) 3. Comparing percentages (Chi Square: χ 2 )

Graphing Data Levels of IV are on horizontal x-axis Levels of IV are on horizontal x-axis DV values are shown on the vertical y-axis DV values are shown on the vertical y-axis y-axis x-axis

Inferential Statistics: Interpreting data

What are inferential statistics? They are a tool used to determine whether or not there is a true relationship between variables or difference between groups They are a tool used to determine whether or not there is a true relationship between variables or difference between groups They are grounded in probability theory. They are grounded in probability theory.

Probability Theory procedures/rules used to predict events procedures/rules used to predict events e.g. regression toward the mean e.g. regression toward the mean True score + random error True score + random error Random error will be responsible for some difference between groups/scores Random error will be responsible for some difference between groups/scores

Inferential Statistics Uses: 1. basic probability theory 2. our knowledge about what things should ‘normally’ look like to figure out if what we observe we could have observed by chance alone

Samples and Populations Samples are a subset of a population that we hope represents the population Samples are a subset of a population that we hope represents the population Inferential statistics help determine how likely it is we would obtain the same result using numerous samples Inferential statistics help determine how likely it is we would obtain the same result using numerous samples

E.g. “95% Confidence Interval” “The president’s approval rating is at 31%, + or – 3 percentage points, with a 95% confidence interval. “The president’s approval rating is at 31%, + or – 3 percentage points, with a 95% confidence interval. takes sample size into account (the bigger the sample, the more representative of the population) takes sample size into account (the bigger the sample, the more representative of the population)

Null and Research Hypotheses Null hypothesis Null hypothesis H o : there is no difference between groups H o : there is no difference between groups Research hypothesis Research hypothesis H 1 : there is a difference between groups H 1 : there is a difference between groups

Null and Research Hypotheses Goal of research is to reject the null hypothesis and accept the research hypothesis Goal of research is to reject the null hypothesis and accept the research hypothesis Null hypothesis is rejected when there is a low probability that the results could be due to random error = statistical significance Null hypothesis is rejected when there is a low probability that the results could be due to random error = statistical significance if we don’t find a statistically significant difference, we ‘fail to reject the null hypothesis’ if we don’t find a statistically significant difference, we ‘fail to reject the null hypothesis’

Probability and Sampling Distributions What is the probability of obtaining the observed results if ONLY random error is operating? What is the probability of obtaining the observed results if ONLY random error is operating? Probability required for significance is called the alpha level Probability required for significance is called the alpha level (e.g..05,.01,.001) If probability is low (.05 or less), reject the null hypothesis If probability is low (.05 or less), reject the null hypothesis If probability is high (over.05), fail to reject the null hypothesis If probability is high (over.05), fail to reject the null hypothesis

Type I and Type II Errors Type I: Made when the null hypothesis is rejected but the null hypothesis is actually true Type II: Made when the null hypothesis is accepted although in the population the research hypothesis is true

What does it mean if results are nonsignificant? could mean that there is no relationship could mean that there is no relationship could be a Type II error could be a Type II error weak manipulation weak manipulation dependent measure not adequate dependent measure not adequate other noise interfered other noise interfered low alpha level low alpha level small sample size small sample size

Correlation Coefficient Numerical index that reflects the relationship between 2 variables Numerical index that reflects the relationship between 2 variables Ranges from –1 to +1 Ranges from –1 to +1 Pearson product-moment correlation or Pearson’s r Pearson product-moment correlation or Pearson’s r

Understanding a correlation Eyeballing your data.8 to 1.0 Very Strong.6 to.8 Strong.4 to.6 Moderate.2 to.4 Weak.0 to.2 Very weak

Scatterplot Illustrates the relationship between variables Illustrates the relationship between variables X on the horizontal axis X on the horizontal axis Y on the vertical axis Y on the vertical axis Positive correlation Positive correlation Data from lower left to upper right Data from lower left to upper right Negative correlation Negative correlation Data from upper right to lower left Data from upper right to lower left

Scatterplot for + correlation

Scatterplot for - correlation

Significance of Pearson r Correlation Coefficient Is the relationship statistically significant? Is the relationship statistically significant? Ho: r = 0 and H1: r 0 Ho: r = 0 and H1: r 0

Importance of Replications Scientists attach little importance to results of a single study Scientists attach little importance to results of a single study Detailed understanding requires numerous studies examining same variables Detailed understanding requires numerous studies examining same variables