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Page 1: Student's T-Test

Student’s t-Test-PIE TUTORS

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Agenda

• Background

• Different versions of t-test

• Main usage of t-test

• t-test v/s z-test

• Assumptions of T-test

• Distribution of t-test and normal distribution

• Case Studies

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Background

• Introduced in 1908 by William Sealy Gosset for the quality control ofbeer.

• Gosset published his mathematical work under the pseudonym“Student”.

• It can be used to determine if two sets of data are significantlydifferent from each other, and is most commonly applied when thetest statistic would follow a normal distribution if the value of ascaling term in the test statistic were known.

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Different Version of T-test

• Single sample t – we have only 1 group; want to test against ahypothetical mean.

• Independent samples t – we have 2 means, 2 groups; no relationbetween groups, e.g., people randomly assigned to a single group.

• Dependent t – we have two means. Either same people in both groups,or people are related, e.g., husband-wife, left hand-right hand, hospitalpatient and visitor.

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Main Usage of t-test

Among the most frequently used t-tests are:

• A one-sample location test of whether the mean of a population has avalue specified in a null hypothesis.

• A two-sample location test of the null hypothesis that the means of twopopulations are equal. All such tests are usually called Student's t-tests,though strictly speaking that name should only be used if the variance ofthe two populations are also assumed to be equal; the form of the testused when this assumption is dropped is sometimes called Welch’s t-test.These tests are often referred to as "unpaired" or "independentsamples" t-tests, as they are typically applied when the statistical unitsunderlying the two samples being compared are non-overlapping.

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Main Usage of t-test

• A test of the null hypothesis that the difference between two responsesmeasured on the same statistical unit has a mean value of zero. Forexample, suppose we measure the size of a cancer patient's tumorbefore and after a treatment. If the treatment is effective, we expect thetumor size for many of the patients to be smaller following thetreatment. This is often referred to as the "paired" or "repeatedmeasures" t-test: see paired difference test.

• A test of whether the slope of a regression line differs significantlyfrom 0.

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Z-test• We can use z-test to test hypotheses about means for large samples

(N>100)

• Consider

• Then

• If

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T-test

We use t-test when the sample size is small (N<100, the usual case) and the population

variance is unknown (the usual case).

Degrees of Freedom

The degrees of freedom for t-distribution are always a simple function of the sample

size, e.g., (N-1).

One way of explaining df is that if we know the total or mean, and all but one score, the

last (N-1) score is not free to vary.

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Assumptions

• Based on assumptions of normality and homogeneity of variance.

• Tested using any statistical package.

• As long as the samples in each group are large and nearly equal, the t-test is robust, that is, still good, even thought assumptions are not met.

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T-Distribution

The t distribution is a short, fat relative of the normal. The shape of t depends on its df. As N becomes infinitely

large, t becomes normal.

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Case StudiesOne Sample t-test

• It is used in measuring whether a sample value significantly differs from a hypothesized value.For example, a research scholar might hypothesize that on an average it takes 3 minutes forpeople to drink a standard cup of coffee. He conducts an experiment and measures how long ittakes his subjects to drink a standard cup of coffee. The one sample t-test measures whetherthe mean amount of time it took the experimental group to complete the task variessignificantly from the hypothesized 3 minutes value.

Paired-Samples t-test• It is used in comparing the means of two variables for a single group. This test computes the

differences between values of two variables for each case and tests whether the average differsfrom 0. For example, in a study on impact of a particular diet on weight, all patients aremeasured at the beginning of the study, prescribed a fixed diet, and measured again. Thus eachsubject has two measures, often called before and after measures.

Independent Samples t-test• The independent-Samples t-test procedure compares means for two groups of cases. Patients

with high blood pressure are randomly assigned to a placebo group and a treatment group. Theplacebo subjects receive an inactive pill, and the treatment subjects receive a new drug that isexpected to lower blood pressure. The two-sample t test is used to compare the average bloodpressures for the placebo group and the treatment group.

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References• http://pic.dhe.ibm.com/infocenter/spssstat/v20r0m0/topic/com.ibm.

spss.statistics.help/idh_ttin.htm

• http://en.wikipedia.org/wiki/Student%27s_t-test

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