Question

Discuss how to identify dependent and independent samples and present examples in your discussion. Further, discuss the hypothesis test procedures for both dependent and independent samples.

Answer 1

• If the values in one sample affect the values in other sample, then the samples are dependent. Dependent samples are generally taken from same population and in pairs.

For eg: Blood sugar level before fasting and after fasting taken from different patients. Since before fasting values and after fasting values have an association, these are dependent samples. Also the observations are taken in pairs and from the sampe population.

• If the values in one sample doesnot affect the values in the other sample, or having no relationship between the samples, then such samples are independent. The independent samples can be taken either from same populatin or from different populations.

For eg: The birth weight of male and female child are taken. Since the birth weight of male child doesnot affect the birth weight of female child , these two samples are independent.

Hypothesis test for dependent samples.(Dependent sample t- test)

This test is applicable only when two samples are not independent and the observations are taken in pairs. In this case, for each observations in one sample, there is a corresponding observation in the other sample. For dependent sample t test, we proceed as follows.

• Let the mean difference in the paired observations in the population be denoted by .
• The null hypothesis is, H0: =0
• Alternative Hypothesis, H1: 0
• Then the test statistic is given by,
  • follows a t distribution with (n-1) degrees of freedom
    • where,
    • 
• While calculating and the sign of the difference is taken into consideration.
• Hence the conclusion is as follows,
  • H0: =0 vs H1: >0, reject H0 if tcal > tn-1()
  • H0:=0 vs H1: <0, reject H0 if tcal < -t​​​​​​n-1()
  • H0: =0 vs H1: 0, reject H0 if tn-1()

​​​​​​Hypothesis test for independent samples.(Independent Sample t-test)

Let the samples be taken from two normal populations and we want to test the hypothesis H0: vs H1:

When the variances of two populations are equal, the test statistic is,

. follows a t- distribution with (m+n-2) degrees of freedom.

where s​​​​​​1​​​​​2 and s​​​22 are the sample variance of two populations.

Then the decisions are given by

H0: Vs , reject H​​​​​​0​​​ if t​​​​cal > t​​​​​m+n-2 ()

H​​​​​​0: Vs H​​​​​​1: , reject H0 if tcal < -tm+n-2

H​​​​​​0: Vs H​​​​​​1: , reject H0 if |tcal| tm+n-2