Stats Ch. 9 T statistic

Describe the numerator in the t formula

The numerator measures the actual difference between the sample data (M) and the population hypothesis (μ).

The shortcoming of using a z-score for hypothesis testing:

is that the z-score formula requires more information than is usually available. Specifically, a z-score requires that we know the value of the population standard deviation (or variance), which is needed to compute the standard error. In most situations, however, the standard deviation for the population is not known.

Note that as the t statistic becomes larger as n becomes ________

larger

Two basic assumptions are necessary for hypothesis tests with the t statistic.

1. The values in the sample must consist of independent observations. The population that is sampled must be normal.

A larger level of confidence (the percentage), produces a _____ t value and a _____ interval.

A larger level of confidence (the percentage), produces a larger t value and a wider interval.

percentage of variance accounted for by the treatment

A measure of effect size that determines what portion of the variability in the scores can be accounted for by the treatment effect.

T-statistic

A statistic used to summarize sample data in situations where the population standard deviation is not known. The t statistic is similar to a z-score for a sample mean, but the t statistic uses an estimate of the standard error.

Under what circumstances is a t statistic used instead of a z-score for a hypothesis test?

A t statistic is used instead of a z-score when the population standard deviation and variance are not known.

Estimated standard error

An estimate of the standard error that uses the sample variance or standard deviation in place of the corresponding population value. Used with the population variance is unknown.

Confidence interval

An interval estimate that is described in terms of the level (percentage) of confidence in the accuracy of the estimation.

True or false. If all other factors are held constant, an 80% confidence interval is wider than a 90% confidence interval.

False.

Some characteristics of a t-distribution

More flat and spread out with a smaller sample size Denominator of the t-statistic formula varies with each sample, whereas with a z-score the denominator is always the same So both numerator and denominator change with each sample for a t-score However, as the degrees of freedom and sample size increase, t distribution looks more normal

t-distribution

The distribution of t statistics is symmetrical and centered at zero like a normal distribution. A t distribution is flatter and more spread out than the normal distribution, but approaches a normal shape as df increases.

Describe the denominator in the t formula

The estimated standard error in the denominator measures how much difference is reasonable to expect between a sample mean and the population mean.

Degrees of freedom

The number of scores in a sample that are independent and free to vary with no restriction. df = n − 1. It represents how well the t statistic represents the z score and the population.

This time the basic rule is as follows: The bigger the sample (n), the _____ the interval.

This time the basic rule is as follows: The bigger the sample (n), the smaller the interval.

True or false. In general, a distribution of t statistics is flatter and more spread out than the standard normal distribution.

True

True or false. If all other factors are held constant, a confidence interval computed from a sample of n = 25 is wider than a confidence interval computed from a sample of n = 100.

True. The smaller sample produces a wider interval

List the magnitude of d for a small effect, medium effect, and large effect

d=0.2 for small effect d=0.5 for medium effect d=0.8 for large effect

3 changes that would decrease the t-value

difference between population and sample mean decreases standard deviation or variance increases sample size decreases

In general, the greater the sample size (n) is, the ___________ the degrees of freedom (n − 1) are, and the better the t distribution approximates the normal distribution

larger

Technically, you do not need the population ______ for a t-test. You just need a null hypothesis, which could be a theory, logical prediction, or wishful thinking

mean

estimated d

mean difference over sample standard deviation

A researcher reports a t statistic with df = 20. How many individuals participated in the study?

n=21

List the magnitude of r squared for a small, medium, and large effect

r=0.01 small effect r=0.09 for medium effect r=0.25 for large effect

3 changes that would increase the t-value

sample mean minus population mean increases standard deviation or variance decreases sample size increases

The only difference between the t formula and the z-score formula is:

that the z-score uses the actual population variance, σ2 (or the standard deviation), and the t formula uses the corresponding sample variance (or standard deviation) when the population value is not known.