T-TEST

Outline  Introduction  T Distribution  Example cases  Test of Means-Single population  Test of difference of Means-Independent Samples  Practical cases on SPSS Rahul Chandra

T-Test  The t-test was introduced in 1908 by William Sealy Gosset, a chemist working for the Guiness Brewery in Dublin, Ireland to monitor the quality of stout – a dark beer.  Because his employer did not want to reveal the fact that it was using statistics for quality control, Gosset published the test in Biometrika using his pen name “Student” (he was a student of Sir Ronald Fisher), and the test involved calculating the value of t. Rahul Chandra

Hypothesis Formulation Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world. Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world. One statement or the other must be true, but they cannot both be true. One statement or the other must be true, but they cannot both be true. H 0 : Null Hypothesis H 1 : Alternative Hypothesis H 0 : Null Hypothesis H 1 : Alternative Hypothesis These two statements are hypotheses because the truth is unknown. These two statements are hypotheses because the truth is unknown. Rahul Chandra

Logic of Hypothesis Testing Efforts will be made to reject the null hypothesis. Efforts will be made to reject the null hypothesis. If H 0 is rejected, we tentatively conclude H 1 to be the case. If H 0 is rejected, we tentatively conclude H 1 to be the case. H 0 is sometimes called the maintained hypothesis. H 0 is sometimes called the maintained hypothesis. H 1 is called the action alternative because action may be required if we reject H 0 in favor of H 1. H 1 is called the action alternative because action may be required if we reject H 0 in favor of H 1. Rahul Chandra

Logic of Hypothesis Testing  Can Hypotheses be Proved? We cannot prove a null hypothesis, we can only fail to reject it. We cannot prove a null hypothesis, we can only fail to reject it.  Role of Evidence The null hypothesis is assumed true and a contradiction is sought. The null hypothesis is assumed true and a contradiction is sought. Rahul Chandra

Sampling distribution  It is the theoretical distribution of an infinite number of samples from the population of interest in your study. It may distribute any sample statistic like mean, sum, proportions etc. For two population it can be of difference of means, etc. Rahul Chandra

Standard error  Every sample always has some inherent level of error, called the standard error. It is basically the standard deviation of sampling distribution. Rahul Chandra

Central Limit Theorem  It states that even if the population distribution is not normal, sampling distribution can still be normal provided the sample size is large enough (> 30).  For small samples normality condition does not hold. Rahul Chandra

T-Distribution  In case the sample size is small (n ≤ 30) and is drawn from a normal population with unknown standard deviation σ, a t test is used to conduct the hypothesis for the test of mean.  The t distribution is a symmetrical distribution just like the normal one.  However, t distribution is higher at the tail and lower at the peak. The t distribution is flatter than the normal distribution.  With an increase in the sample size (and hence degrees of freedom), t distribution loses its flatness and approaches the normal distribution whenever n > 30. Rahul Chandra

T-Distribution  A comparative shape of t and normal distribution is given in the figure below: Rahul Chandra

Assumptions Rahul Chandra

Test Concerning Means – Case of Single Population Case of large sample - In case the sample size n is large or small but the value of the population standard deviation is known, a Z test is appropriate. The test statistic is given by, Rahul Chandra

Testing a Mean: Known Population Variance Step 1: State the hypotheses For example, H 0 :  216 Step 1: State the hypotheses For example, H 0 :  216 Step 2: Specify the decision rule For example, for  =.05 for the right-tail area, Reject H 0 if z > 1.645, otherwise do not reject H 0 Step 2: Specify the decision rule For example, for  =.05 for the right-tail area, Reject H 0 if z > 1.645, otherwise do not reject H 0 Rahul Chandra

Testing a Mean: Known Population Variance  For a two-tailed test, we split the risk of Type I error by putting  /2 in each tail. For example, for  =.05 Rahul Chandra

Test Concerning Means – Case of Single Population If the population standard deviation σ is unknown, the sample standard deviation is used as an estimate of σ. There can be alternate cases of two-tailed and one-tailed tests of hypotheses. Corresponding to the null hypothesis H 0 : μ = μ 0, the following criteria could be formulated as shown in the table below: Rahul Chandra

Test Concerning Means – Case of Single Population The null hypothesis to be tested is: H 0 : μ = μ 0 The alternative hypothesis could be one-tailed or two-tailed test. The test statistics used in this case is: The procedure for testing the hypothesis of a mean is identical to the case of large sample. Rahul Chandra

Tests for Difference Between Two Population Means Case of large sample - In case both the sample sizes are greater than 30, a Z test is used. The hypothesis to be tested may be written as: H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 Where, μ 1 = mean of population 1 μ 2 = mean of population 2 The above is a case of two-tailed test. The test statistic used is on the Rahul Chandra

Tests for Difference Between Two Population Means The Z value for the problem can be computed using the above formula and compared with the table value to either accept or reject the hypothesis. Rahul Chandra

Tests for Difference Between Two Population Means Case of small sample - If the size of both the samples is less than 30 and the population standard deviation is unknown, the procedure described above to discuss the equality of two population means is not applicable in the sense that a t test would be applicable under the assumptions: a) Two population variances are equal. b) Two population variances are not equal. Rahul Chandra

Tests for Difference Between Two Population Means Population variances are equal - If the two population variances are equal, it implies that their respective unbiased estimates are also equal. In such a case, the expression becomes: To get an estimate of σˆ 2, a weighted average of s 1 2 and s 2 2 is used, where the weights are the number of degrees of freedom of each sample. The weighted average is called a ‘pooled estimate’ of σ 2. This pooled estimate is given by the expression: Rahul Chandra

Tests for Difference Between Two Population Means The testing procedure could be explained as under: H0 : μ1 = μ2 ⇒ μ1 – μ2 = 0 H1 : μ1 ≠ μ2 ⇒ μ1 – μ2 ≠ 0 In this case, the test statistic t is given by the expression: Once the value of t statistic is computed from the sample data, it is compared with the tabulated value at a level of significance α to arrive at a decision regarding the acceptance or rejection of hypothesis. Rahul Chandra

Tests for Difference Between Two Population Means Population variances are not equal - In case population variances are not equal, the test statistic for testing the equality of two population means when the size of samples are small is given by: The degrees of freedom in such a case is given by the expression: The procedure for testing of hypothesis remains the same as was discussed when the variances of two populations were assumed to be same. Rahul Chandra