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Chapter 9: the t statistic - Coggle Diagram
Chapter 9: the t statistic
T statistic. Used for hypothesis testing when the population standard deviation is not known.
In most cases, the population standard deviation is not known, so a t-test is used more often than the z-score
T statistic equation. T = M - µ / sᴹ. Similar to z-score formula except it uses the estimated standard error of M in the formula.
T equals the sample mean - population mean / estimated standard error of M. A t test can be used when the population mean is a logical prediction or wishful thinking.
Estimated standard error of M. Symbol is s / √n. s is the standard deviation of the sample. Provides an estimate of distance between the sample mean and population mean.
Equation is s / √n or √s²/n for estimated standard error of M.
Degrees of freedom. Describes the number of scores in a sample that are independent and free to vary.
The greater the sample size (n), the greater the degrees of freedom (n-1), the better the t distribution approximates the normal distribution.
A t distribution is the complete set of t values calculated for all samples of size (n) or degrees of freedom (n-1). It approximates the shape of a normal distribution.
When the degrees of freedom are small (ex. 5), then the t distribution is flatter than a normal distribution.
As sample size and df increase, the variability in the t distribution decreases.
Hypothesis testing example using a t statistic
State the hypothesis and select the alpha level. Ex. Scores on a test before intervention is 50. Alpha level is .05 for two
Locate the critical region. df = n - 1. Ex. 9 test takers is 9-1=8. Use table to find critical region for alpha level .05 for two tailed test with 8 degrees of freedom.
Calculate the test statistic. Find sample variance s² = SS/n-1. Equals 162/8 =20.25. Then find the estimated standard error of M = √s²/n = √20.25/9 =√2.25 =1.5.
Compute the sample statistic t. t = M - µ / Sm. Example if M = 54, then (54 - 50) / 1.5 = 2.67
Make a decision about null hypothesis. A t score of 2.67 falls in the critical region outside of the alpha level of .05, so we reject the null hypothesis.
Assumption 1 of t-test. Values of the sample must have independent observations.
Assumption 1 of t-test. The population sampled must be normal.
Implications of the t-statistic. The larger the estimated standard error, the more likely the t statistic will be closer to zero and thus reduces the likelihood of rejecting the null hypothesis.
If the variance of the sample is high, then the standard error will be high. If the sample size is large, then the standard error will be low.
Cohen's d is the mean difference / sample standard deviation.
r-squared. The percentage of variance explained by the treatment. Explained sum of squares over the total sum of squares.
r-squared = t² / (t² + df). R-squared .01 is little effect, .09 is medium effect, .25 is large effect.
Confidence interval. Range of values centered around a sample statistic. Ex. 95% confidence interval of a sample mean
To calculate the confidence interval, find the t-value and degrees of freedom. Ex. 8 degrees of freedom, the t value is +/-2.306 for a 95% confidence interval.
A confidence interval is expressed by the equation µ = M +/- t (Sm). It expresses an interval around the sample mean to estimate the population mean.
If you set the confidence interval at 99%, then you will have a large width, so standard practice is 95%.
If you have a larger sample size, then confidence width will be smaller as well.