Chapter 9 - One-sample T-test

What assumptions are made about the one-sample t-test?

One underlying assumption of the one-sample t-test is that the sample is independently and randomly selected from the population of interest. The other underlying assumption is that the scores on the variable of interest are normally distributed in the population. This also makes sure that the incidence of Type I errors will be equal to alpha and the incidence of Type II errors will be equal to beta.

What are the degrees of freedom for a one-sample t-test?

n-1

When do you use a t-test instead of a z-test?

A z-score is used when the population standard deviation or variance is known. The t-statistic is used when the population standard deviation or variance is unknown.

How does increasing sample size influence the shape of the t-distribution?

As, the sample size, n, increases, the shape of distribution approaches to a normal distribution (bell curve), because, standard error of the mean decreases with the increase of n. This means that the area near the center increases while the area near the tails decreases. When df decreases, the area near the center decreases while the area near the tails increases. So when the sample size is bigger (more degrees of freedom) the t-distribution is more peaked, less spread out and a smaller difference between means or between a mean and a criterion value before we are willing to say that it is likely a real difference.

What does effect size (r2) mean? What is the percentage of variance explained by the treatment?

Effect size is a statistical concept that measures the strength of the relationship between two variables on a numeric scale. For instance, if we have data on the height of men and women and we notice that, on average, men are taller than women, the difference between the height of men and the height of women is known as the effect size. The greater the effect size, the greater the height difference between men and women will be.

What is the influence of sample size and variance on the t - statistic?

The smaller the sample size, the greater the influence of the values of individual samples on variance. This variability becomes stable as the sample size increases. If the sample sizes of the groups are different, then this difference in variability may result in different variances.

What is the shape of the t distribution?

The t distribution is very similar to the normal distribution when the estimate of variance is based on many degrees of freedom, but has relatively more scores in its tails when there are fewer degrees of freedom. Notice that the normal distribution has relatively more scores in the center of the distribution and the t distribution has relatively more in the tails. The shape of t-distribution approaches as sample size increases.

Hypothesis testing using a one-sample t-test.

There are two kinds of hypotheses for a one-sample t-test, the null hypothesis, and the alternative hypothesis. The alternative hypothesis assumes that some difference exists between the true mean (μ) and the comparison value (m0), whereas the null hypothesis assumes that no difference exists. The purpose of the one-sample t-test is to determine if the null hypothesis should be rejected, given the sample data.

How do you look up critical values in the t-table?

To find a critical value, look up your confidence level in the bottom row of the table; this tells you which column of the t-table you need. Intersect this column with the row for your df (degrees of freedom). The number you see is the critical value (or the t*-value) for your confidence interval.