Chapter 9: The Single Sample t Test and the Paired Samples t Test

single sample t test

hypothesis test in which we compare a sample from which we collect data to a population for which we know the mean but not the standard deviation

paired samples t test

hypothesis test that is used to compare two means for a within-groups design, a situation in which every participant is in both samples = AKA dependent samples t test

confidence intervals for a single sample t test

here are the steps for a confidence interval using a single sample t test = 1) draw a picture of a t distribution that includes the confidence interval = 2) indicate the bounds of the confidence interval on the drawing = 3) look up the t statistic that fall at each line marking the middle 95% (for a 95% confidence interval) = 4) convert the t statistics back into raw means -- M(lower) = -t(s(M)) + M(sample) -- M(upper) = t(s(M)) + M(sample) = 5) verify that the confidence interval makes sense

six steps for a paired samples t test

here are the steps for a paired samples t test = 1) identify the populations, distribution, and assumptions -- in this case, the comparison distribution is a distribution of mean difference scores (paired samples t test) = 2) state the null and alternative hypotheses = 3) determine the characteristics of the comparison distribution -- in this case, the mean is going to be 0, since there is supposed to be no mean difference -- for the standard error, take the individual scores in the sample and find the differences, then square each individual difference ~ after this, calculate the sum of squares, then put into the standard error formula of the single sample t test = 4) determine the critical values, or cutoffs -- for degrees of freedom, this is the number of participants, not the number of scores, minus one = 5) calculate the test statistic -- for the expected mean, use the mean of the differences of the scores to calculate the test statistic = 6) make a decision

effect size for a single sample t test

the same conventions apply for this test as for a z test = small effect = 0.2 = medium effect = 0.5 = large effect = 0.8 formula = Cohen's d = (M - mu)/s when working with APA, the report for d goes like this .... there is an effect, t(4) = 2.87, p<0.05, d = 1.29.

confidence interval for a paired samples t test

here are the steps for a paired samples t test confidence interval = 1) draw a picture of a t distribution that includes the confidence interval - use the sample mean at the center of the distribution, not the population distribution of 0 = 2) indicate the bounds of the confidence interval on the drawing = 3) add the critical t statistics to the curve = 4) convert the critical t statistics back into raw mean differences - same formulas as used in the single sample t test = 5) verify that the confidence interval makes sense NOTE: for effect size, d, the formula and reporting system for the APA is identical to single sample t tests

the steps of a single sample t test

here are the steps to conducting a single sample t test steps = 1) identify the populations, distribution, and assumptions -- comparison distribution is a distribution of means (single sample t test) -- does not have to meet all 3 requirements, but must at least meet the first - dependent variable is scale - participants are randomly selected - data is normally distributed, or has a n = 30 = 2) state the null and alternative hypotheses - Ho: mu(1) = mu(2) - Ha: mu(1) =/= mu(2) = 3) determine the characteristics of the comparison distribution - calculate the sample mean and the standard error for the t statistic = 4) determine the critical values, or cutoffs - use df before consulting the t table - if a df is too large or in the middle of values on a t table, use the smaller df, or the more conservative critical value (the larger one) = 5) calculate the test statistic = 6) make a decision - if the test statistic is a greater value than the critical value, then you must reject the null hypothesis - if the test statistic is a lesser value than the critical value, you must fail to reject the null hypothesis

APA style for reporting statistics

here are the steps to reporting the results of a single sample t test = write the symbol for the test statistic (t) = write the degrees of freedom in parentheses = write an equal sign and then the value of the test statistic, typically to two decimal places = write a comma and then indicate the p-value by writing p = and then the actual value -- we will not know the actual value of p, so just indicate if the p-value is beyond the critical value by saying p<0.05 when we reject the null hypothesis or p>0.05 when we fail to reject the null hypothesis = example of the finished product - t(4) = 2.87, p<0.05 = report would also include a sample mean and standard deviation (not standard error) rounded to two decimal points - (M = 7.80, SD = 2.49)

standard deviation for a t test

similar to a standard deviation for a z test, but must divide by n - 1 formula = s = sqrt(E(X - M)^2/(N - 1)) = this formula is used for estimating the population standard deviation = steps -- calculate the sample mean -- use the sample mean in the corrected formula for the standard deviation

t statistic

statistic that indicates the distance of a sample mean from a population mean in terms of the estimated standard error formula = t = (M - mu(M))/s(M) = the t statistic is a more conservative estimate than the z statistic

degrees of freedom

the number of scores that are free to vary when we estimate a population parameter from a sample formula = df = N - 1 = on a t table, as df go up, critical values go down = the level of confidence in the observations increases as the critical value decreases

distribution of mean differences

these are the steps for how to construct a distribution of mean differences = randomly choose three pairs of cards, replacing each pair of cards before randomly selecting the next = for each pair, calculate a difference score by subtracting the first weight from the second weight = calculate the mean of the differences in weights for these three people -- complete the three steps again -- randomly choose three people from the population of many college students, calculate the difference scores, and calculate the mean of the three difference scores

t distribution

this distribution is used for: = when you don't know the population standard deviation = when you compare two samples

standard error for the t statistic

this is the spread of the distribution of means formula = s(m) = s/sqrt(N) = distribution of means is less variable than a distribution of scores = we use the corrected (s) rather than the population (omega) standard deviation = by the CLT, the value of the standard error is smaller than the value for the standard deviation

n affect on the t distribution

when n is small, the t distribution is flatter and more spread out = looks more like the z distribution when the sample size n is large