T-test

T-test for one and two population means

1. Analysis with t-Test T-Test for One Population mean and Two Population means

2. Background Introduced in 1908 by William Sealy Gosset for the quality control of Beer. Gosset published his mathematical work under the pseudonym “Student”.

3. The t-test assumes : Types of t-test: There are two main types of t-test : Independent-measures t-test : when samples are not matched. Matched-pair t-test : When samples appear in pairs ( eg . before-and-after). It is usually used with small (<30) samples that are normally distributed .

4. One Sample t-Test One sample  t -test is a statistical procedure used to examine the mean difference between the sample and the known value of the population mean. In one sample t-test, we know the population mean .  We draw a random sample from the population and then compare the sample mean with the population mean and make a statistical decision as to whether or not the sample mean is different from the population mean.

5. The Assumptions of One-Sample t-Test The data are continuous. The data follow the Normal distribution. The sample is a simple random sample from the population.

6. Hypotheses and Formulas With

7. Example of ONE-Sample t-Test W hen we take a sample from the city and we know the mean of the country (population mean).  If we want to know whether the city mean differs from the country mean, we will use the one sample  t -test

8. Two Samples /Independent Samples t-test Equal Variances The independent two-sample t-test is used to test whether population means are significantly different from each other, using the means from randomly drawn samples . Any statistical test that uses two samples drawn independently of each other and using t-distribution, can be called a 'two-sample t-test'.

9. The assumptions of Independent Sample t-Test The data are continuous The data follow the Normal distribution. The variances of the two populations are equal The two samples are independent Both samples are simple random samples from their respective populations.

10. Hypotheses and Formulas using POOLED variences With

11. Example Suppose that a school has two buildings - one for girls and the other for boys. Suppose that the principal want to know if the pupils of the two buildings are working equally hard, in the sense that they put in equal number of hours in studies on the average. Statistically speaking, the principal is interested in testing whether the average number of hours studied by boys is  significantly different  from the average for girls.

12. Test name Situation Data type One sample t-test We compare a group with standard value Quantitative Independent sample t-test We compare the averages of two independent groups Quantitative Summary