Chapter 9: Introduction to the t Statistic
An alternative to z
Is used to test hypotheses about an unknown population mean μ, when the value of s is unknown.
estimated standard error is in the denominator.
Estimated Standard Error (sM): used as an estimate of the actual standard error, σM, when the value of s is unknown. It's computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean M and the population mean μ.
Degrees of freedom, df (n-1): describes the number of scores in a sample that are independent and free to vary.
t distribution: the complete set of t values computed for every possible random sample for a specific sample size (n) or a specific degrees of freedom df. t distribution approximates the shape of a normal distribution.
Shape of t distribution: exact shape of t distribution changes with degrees of freedom.
As df gets larger, the t distribution gets closer in shape to a normal z-score distribution.
Uses a t distribution table to find proportions for "t* statistics.
t: sample mean(from the data) - population mean(hypothesized from H0 / estimated standard error(computed from the sample data)
zort: actual difference between sample (M)and the hypothesis (μ) / expected difference betweenMandμwith no treatment effect
known population before treatment with mean ----> unknown population after treatment with no mean.... conducting hypothesis testing to determine whether the treatment has any effect.
Assumptions of the t test: 1. The values in the sample must consist of independent observations. 2. The population sampled must be normal.
Sample size and sample variance have a big effect on the t statistic. Any factor that influences the standard error also effects the likelihood of rejecting the null hypothesis and finding a significant treatment effect.
Estimated Cohen's d formula: mean difference/standard deviation = μtreatment-μno treatment/σ
------> since the population mean with treatment and standard deviation is unknown, we then use the mean for the treated sample and the standard deviation for the sample after treatment as estimates of the unknown parameters with a new formula of estimatedd: mean difference/sample standard deviation = M-μ/s
Alternative method for measuring effect size deals with variability in the scores which is explained by the treatment effect.
Confidence interval: an interval, or range of values centered around a sample statistic.
Two factors that affect the width of a confidence interval are the change in the level of confidence and sample size.
Directional (one-tailed) test is used in some research situations such as exploratory investigations, pilot studies, or priori justification.
Critical region for a one-tailed test: process can help that can eliminate the need to determine which tail (left or right) should contain the critical region
Critical region process divided into 2 stages: Stage 1 is simply to determine whether the sample mean is in the direction predicted by the original research question. Stage 2 is to determine whether the effect is large enough to be significant.