Exam 3 Stats

How is the calculation of standard error different for the t test than for the z test?

For both tests, standard error is calculated as the standard deviation divided by the square root of N. For the z test, the population standard deviation is calculated with N in the denominator. For the t test, the standard deviation for the population is estimated by dividing the sum of squared deviations by N-1.

t distributions

Help us specify how confident we can be in our research findings. Tells us if the sample differs from the larger population.

In what situation do we conduct a paired-samples t test? In what situation do we conduct an independent samples t test?

When the data we are comparing were collected using the same participants in both conditions, a paired-samples t test is used; each participant contributes two values to the analysis. When we are comparing two independent groups and no participant is in more than one condition, we use an independent samples t test.

When do we use a paired-samples t test?

When we are comparing two groups and the same people are in both groups.

When is it appropriate to use a single-sample t test?

When we want to compare a sample to a population but we do not know the population's standard deviation.

How does considering the conclusion in terms of effect size help to prevent incorrect interpretations of the findings?

Effect size tells us how large or small the difference we observed is, regardless of sample size. Even when a result is statistically significant, it might not be important. Effect size helps us evaluate practical significance.

How do we conduct a paired-samples t test?

For a paired-samples t test we calculate a difference score for every individual. We then compare the average difference observed to the average difference we would expect based on the null hypothesis. If there is no difference, then all difference scores should average to 0.

Why do we calculate confidence intervals?

Confidence intervals add details to the hypothesis test. Specifically, they tell us a range within which the population mean would fall 95% of the time if we were to conduct repeated tests using samples of the same size from the same population.

How is the critical t value affected by sample size and degrees of freedom?

Critical t values decrease in magnitude as sample size increases.

Why is a single-sample t test more useful than a z test?

A single-sample t test has more uses than a z test because we only need to know the population mean (not the population standard deviation) for the single-sample t test.

When is it appropriate to use the independent-samples t test?

An independent samples t test is used when we do not know the population parameters and we are comparing two groups that are composed of nonoverlapping, unrelated participants or observations.

Explain what an individual score is, as it is used in a paired-samples t-test.

An individual difference score is a calculation of change or difference for each participant. For example, we might subtract weight before the holiday break from weight after the break to evaluate how many pounds an individual lost or gained.

Confidence Interval

An interval estimate based on the sample statistic; it includes the population mean a certain percentage of the time if we sample from the same population repeatedly.

Why do the t distributions merge with the z distributions as sample size increases?

As the sample size increases, we can feel more confident in the estimate of the variability in the population. Remember, this estimate of the variability (s) is calculated with N-1 in the denominator in order to inflate the estimate somewhat. As the sample increases from 10 to 100 for example, and then up to 1000, subtracting 1 from N has less of an on the overall calculation. As this happens, the t distribution approach the z distribution, where we in fact knew the population standard deviation and did not need to estimate it.

Interval Estimate

Based on a sample statistic and provides a range of plausible values for the population parameter.

Why is the population mean almost always equal to 0 for the null hypothesis in the two-tailed paired-samples t test?

Because the null hypothesis is that there will not be a difference between the two conditions.

How is a paired-samples t test similar to a single-sample t test?

Both involve comparing a sample mean to a population mean, but a paired-sample t test uses a mean difference and the population mean is typically zero.

What effect does increasing the sample size have on standard error and the test statistic?

Increasing the sample size makes the standard error smaller which in turn makes the test statistic larger (more extreme).

What is a t statistic?

Indicates the distance of a sample mean from a population mean in terms of the estimated standard error.

Why are interval estimates better than point estimates?

Interval estimates provide a range of scores in which we have some confidence the population statistic will fall, whereas point estimates use just a single value to describe the population.

Degrees of freedom

Is the number of scores that are free to vary when estimating a population parameter from a sample.

How does creating a confidence interval for a paired-samples t test give us the same information as hypothesis testing with a paired-samples test?

The null hypothesis for the paired samples t test is that the mean difference score is 0, that is um = 0. Therefore, if the confidence interval around the mean difference does not include 0 we know that the sample mean is unlikely to have come from a distribution with a mean of 0 and we can reject the null hypothesis.

When should we use the t distribution?

The t distributions are used when we do not know the population standard deviation and we are comparing two groups.

In statistics, why don't we talk about having 100% confidence?

The tails of a normal curve extend to infinity, so we can never include 100% of the possible values while still providing helpful information.

Explain the distinction between the terms independent samples and paired samples as they relate to t tests?

The term paired = samples is used to describe a test that compares an individuals scores in two conditions; it is also called a paired-samples t test. The term independent = samples refers to groups that do not overlap in any way, including membership; the observations made in one group in no way relates to or depends on the observation made in another group.

What specific danger exists when reporting a statistically significant difference between two means?

There may be a statistically significant difference between group means, but the difference might not be meaningful or have a real-life application.

How is a pared-samples t test difference from a single-sample t test?

Unlike a single sample t test, in the paired samples t test we have two scores for every participant; we take the difference between these scores before calculating the sample mean difference that will be used in the t test.

Paired-samples t test

Used to compare two means for a within-groups design, a situation in which every participant is in both samples; also called a dependent samples t test.

As measures of variability, what is the difference between standard deviation and variance?

Variance is measured in squared unites, whereas standard deviation is the square root of the variance, so it is measured in the original units.

Why would we want the variability estimate based on a larger sample to count more (to be more heavily weighted) than one based on a smaller sample?

We assume larger populations do a better job of estimating the population than smaller samples do, so we would want the variability measure based on the larger sample to count more.

Why do we calculate confidence intervals?

We calculate confidence intervals to determine a range of plausible values for the population parameter, based on the data.