Types of T-Tests

8/10/2019 Types of T-Tests

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Types of t-tests

A t-test is a hypothesis test of the mean of one or two normally distributed populations. Several

types of t-tests exist for different situations, but they all use a test statistic that follows a t-

distribution under the null hypothesis:

Test Purpose Example

1 sample t-test Tests whether the mean of a single

population is equal to a target value

Is the mean height of female college students greater

than 5.5 feet?

2 sample t-test Tests whether the difference

between the means of two

independent populations is equal to

a target value

Does the mean height of female college students

significantly differ from the mean height of male

college students?

paired t-test Tests whether the mean of the

differences between dependent or

paired observations is equal to a

target value

If you measure the weight of male college students

before and after each subject takes a weight-loss pill, is

the mean weight loss significant enough to conclude

that the pill works?

t-test in

regression output

Tests whether the values of

coefficients in the regression

equation differ significantly from

zero

Are high school SAT test scores significant predictors

of college GPA?

An important property of the t-test is its robustness against assumptions of population normality. In

other words, t-tests are often valid even when the assumption of normality is violated, but only if

the distribution is not highly skewed. This property makes them one of the most useful procedures

for making inferences about population means.

However, with nonnormal and highly skewed distributions, it might be more appropriate to use

nonparametric tests.


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Why should I use a 1-sample t-test?

To perform this test, choose:

Mac: Statistics> 1-Sample Inference> t

PC: STATISTICS> One Sample> t

Use a 1-sample t-test to estimate the mean of a population and compare it to a target or reference

value when you do not know the standard deviation of the population. Using this test, you can do

the following:

Determine whether the population mean differs from the hypothesized mean that you

specify.

Calculate a range of values that is likely to include the population mean.

For example, the manager of a pizza business collects a random sample of pizza delivery times. The

manager uses the 1-sample t-test to determine whether the mean delivery time is significantly lower

than a competitor's advertised delivery time of 30 minutes.

The test calculates the difference between your sample mean and the hypothesized mean relative to

the variability of your sample. Usually, the larger the difference and the smaller the variability in

your sample, the greater the chance that the population mean differs significantly from the

hypothesized mean.

The 1-sample t-test also works well when the assumption of normality is violated, but only if the

underlying distribution is symmetric, unimodal, and continuous. If the values are highly skewed, it

might be appropriate to use a nonparametric procedure, such as a 1-sample sign test.

For 1-sample t, the hypotheses are:

Null hypothesis

H0: = 0 The population mean () equals the hypothesized mean (0).

Alternative hypothesis

Choose one:

H1: 0 The population mean () differs from the hypothesized mean (0).

H1: > 0 The population mean () is greater than the hypothesized mean (0).

H1: < 0 The population mean () is less than the hypothesized mean (0).


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Why should I use a 2-sample t test?


Mac: Statistics> 2-Sample Inference> t

PC: STATISTICS> Two Samples> t

Use a 2-sample t-test to do the following:

Determine whether the population means of two independent groups differ.

Calculate a range of values that is likely to include the difference between the population

means.

For example, you want to determine whether two grain dispensers are dispensing the same amount

of grain.

2-Sample t calculates a confidence interval and does a hypothesis test of the difference between two

population means when standard deviations are unknown and samples are drawn independentlyfrom each other. This procedure is based on the t-distribution, and for small samples it works best if

the data were drawn from distributions that are normal or close to normal. You can have increasing

confidence in the results as the sample sizes increase.

To do a 2-sample t-test, the two populations must be independent; in other words, the observations

from the first sample must not have any bearing on the observations from the second sample. For

example, test scores of two separate groups of students are independent, but before-and-after

measurements on the same group of students are not independent, although both of these examples

have two samples. If you cannot support the assumption of sample independence, reconstruct your

experiment to use the paired t-test for dependent populations.

The 2-sample t-test also works well when the assumption of normality is violated, but only if the

underlying distribution is not highly skewed. With nonnormal and highly skewed distributions, it

might be more appropriate to use a nonparametric test.

For 2-sample t, the hypotheses are:

Null hypothesis

H0: 12= 0 The difference between the population means (12) equals the

hypothesized difference (0).


Choose one:

H1: 120 The difference between the population means (12) does not equal the


H1: 12> 0 The difference between the population means (12) is greater than the



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H1: 120 The difference between the population means (12) does not equal the


H1: 12< 0 The difference between the population means (12) is less than the


Why should I use a paired t test?


Mac: Statistics> 2-Sample Inference> Paired t

PC: STATISTICS> Two Samples> Paired t

Use a paired t-test to do the following:

Determine whether the mean of the differences between two paired samples differs from 0.

Calculate a range of values that is likely to include the population mean of the differences.

Use this analysis to:

Determine whether the mean of the differences between two paired samples differs from 0

(or a target value)

Calculate a range of values that is likely to include the population mean of the differences

For example, suppose managers at a fitness facility want to determine whether their weight-loss

program is effective. Because the "before" and "after" samples measure the same subjects, a paired

t-test is the most appropriate analysis.

The paired t-test calculates the difference within each before-and-after pair of measurements,

determines the mean of these changes, and reports whether this mean of the differences is

statistically significant.

A paired t-test can be more powerful than a 2-sample t-test because the latter includes additional

variation occurring from the independence of the observations. A paired t-test is not subject to this

variation because the paired observations are dependent. Also, a paired t-test does not require both

samples to have equal variance. Therefore, if you can logically address your research question with

a paired design, it may be advantageous to do so, in conjunction with a paired t-test, to get more

statistical power.

The paired t-test also works well when the assumption of normality is violated, but only if the

underlying distribution is symmetric, unimodal, and continuous. If the values are highly skewed, it

might be appropriate to use a nonparametric procedure, such as a 1-sample sign test.

For paired t, the hypotheses are:

Null hypothesis


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H0: d= 0 The population mean of the differences (d) equals the

hypothesized mean of the differences (0).


Choose one:

H1: d 0 The population mean of the differences (d) does not equal the

hypothesized mean of the differences (0).

H1: d> 0 The population mean of the differences (d) is greater than the hypothesized

mean of the differences (0).

H1: d< 0 The population mean of the differences (d) is less than the hypothesized

mean of the differences (0).

Types of T-Tests

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