Question

Which of the following research is most likely to use an independent t test?

Answer 1

The independent-t test compares the means of two indpendent groups inorder to determine whether there is statistical evidence that the associated population means are significantly different. This is a parametric test.

It uses two separate samples to evaluate the mean difference between two treatments or between two populations.

For Example:

​ Comparing the blood pressure readings before medication and after medication for a group of patients with high blood pressure.

HYPOTHESIS:

​​ ​(i.e) The two population means are equal.

​ (i.e) The two population means are not equal.

   (OR)

​ (i.e) The difference between two population means are equal.

(i.e) The difference between two population means are not equal.

where and ​ are population means for group1 and group2.

VARIABLES USED IN THIS TEST:

• ​Dependent or test variable
• Independent or grouping variable

HOMOGENEITY OF VARIANCE ASSUMPTION:

​ One notable assumption for independent t-test is homogeneity of variance (i.e) the two populations from which tha samples are obtained have equal variances. The test used is called Levene's test.

The hypothesis is given by:

​ (i.e) the population variance of group 1 and 2 are equal.

(i.e) the population variance of group 1 and 2 are not equal.

TWO FORMS OF TEST STATISTIC:

​ There are two forms of test statistic for independent t-test based on whether or not equal variances are assumed.

(i) EQUAL VARIANCES ASSUMED:

When two independent samples are asumed to be drawn from population with identical population variances the test statistic is,

  

where

x1bar-Mean of first sample

x2bar-Mean of second sample

n1-Sample size of first sample

n2-Sample size of second sample

s1-Standard deviation of first sample

s2-Standard deviation of second sample

sp-Pooled standard deviation

(ii) EQUAL VARIANCES NOT ASSUMED:

When two independent samples are asumed to be drawn from population with identical population variances ​ the test statistic is,​

where

x1bar-Mean of first sample

x2bar-Mean of second sample

n1-Sample size of first sample

n2-Sample size of second sample

s1-Standard deviation of first sample

s2-Standard deviation of second sample

COMMON USES:

• Statistical difference between the means of two groups
• Statistical difference between the means of two interventions
• Statistical difference between the means of two change scores

OTHER USES:

​​​​1. The independent t-test is also used to combine the variances from two separate samples to obtain one value for "pooled variance" ​. The pooled variance is an average of two sample variances but is computed so that the large sample carries more weight if the samples are not the same size.

For Example: If one sample has the variance of 20 and second sample has a variance of 30 then the pooled variance for these two samples is somewhere between 20 and 30.

2. The independent t test combines the error for each of the two sample means to obtain one standard error for the mean difference.

For Example:If two samples each with n=6 subjects, produce a pooled variance of 20, then the estoimated standard error for the mean difference is the square root of [(20/6)+(20/6)].

3. In addition to a hypothesis test evaluating the mean difference, the independent t statistic can also be used to construct a confidence interval estimating the mean difference between two treatments.