One Sample T Tests
ANOTHER MEASURE OF EFFECT SIZE: CONFIDENCE INTERVALS
- Our best estimates of the population values are obtained from our sample values (e.g, M is our best guess of ). •But these estimates are subject to error, and the size of that error is determined by sample size (law of large numbers). •The central limits theorem tells us that our estimate is determined by the standard error of the mean •We compute a range for our mean estimate by converting our scores to a known probability distribution, like z or t.
ANOTHER MEASURE OF EFFECT SIZE: r2
- r2 measures the proportion of variance in the data that is accounted for by the treatment effect. total variance in the data = treatment effect + unexplained variance(error) - Based on the idea that the treatment causes the scores to change, which contributes to the observed variability in the data. • BUT some variability due to sampling error
Steps for Hypothesis testing with t
1. State H0, H1, and choose 2. Determine what type of observation it would take to reject H0 a. What is the appropriate test statistic? b. What is the critical value? (calculate df, lookup t) 3. Evaluate the sample data a. Calculate sample M and s (if needed) b. Calculate the standard error c. Calculate t 4. Reach a conclusion
ASSUMPTIONS OF THE t TEST
1. The values in the sample must consist of independent observations. 2. The population sampled must be normal.
PROBLEMS WITH COHEN'S D
Seems arbitrary and hard to interpret
CONFIDENCE INTERVALS
Steps in constructing confidence intervals: 1. Determine interval size (e.g., 95%) 2. Look up appropriate statistic, using the two-tailed value (e.g., t score) 3. Confidence interval is given by: CI = M +/- t*(standard error)
HYPOTHESIS TESTING WITH t- TEST
The probabilities associated with t-values try to correct for the error in estimating . As a result, it is typically harder to reject the null hypothesis with t than with z.
WHEN TO USE A t TEST
When the ø is unknown
MEASURE OF EFFECT SIZE: COHEN'S D
small (< .2 SD), medium ( ≈.5 SD), large (> .8 SD)
t Distribution
t distribution is normal when n is large, but flatter than normal for small values of n
THE INFLUENCE OF SAMPLE SIZE AND SAMPLE VARIANCE
•Sample size and variance both have a large effect on the t-statistic and thereby influence the statistical decision. •Estimated standard error is inversely related to the number of scores in the sample •Large n and small s leads to larger t. More likely to reject H0