Given two dependent random samples with the following results:

Population 1 52 81 72 67 73 73 76 83
Population 2 61 71 82 62 79 77 71 85
Can it be concluded, from this data, that there is a significant difference between the two population means? Let d = d (Population 1 entry) − (Population 2 entry). Use a significance level of α = 0.01 for the test. Assume that both populations are normally distributed.

Step 1 of 5: State the null and alternative hypotheses for the test.
Step 2 of 5: Find the value of the standard deviation of the paired differences. Round your answer to one decimal place.
Step 3 of 5: Compute the value of the test statistic. Round your answer to three decimal places.
Step 4 of 5: Determine the decision rule for rejecting the null hypothesis H0. Round the numerical portion of your answer to three decimal places.
Step 5 of 5: Make the decision for the hypothesis test. Reject or Fail to regect Null Hypotthesis.

We Fail to reject Null Hypothesis.

Step-by-step explanation:

We are given two dependent random samples;

Population 1   Population 2   d = Population 1 entry-Population 2 entry    

      52                       61                                52 - 61 = -9                                 81

      81                        71                                 81 - 71 = 10                                 100

      72                        82                                72 - 82 = -10                             100

      67                        62                                67 - 62 = 5                                25

      73                        79                                 73 - 79 = -6                               36

      73                        77                                 73 - 77 = -4                                16

      76                        71                                  76 - 71 = 5                                 25

      83                        85                                 83 - 85 = -2                               4    

                                                                                                              ∑= 387                                  

Now,  Null Hypothesis, :    or  

 Alternate Hypothesis, :     or  

The test statistics for paired data is given by;

                    follows

 where, = Mean of Population 2 data - Mean of Population 1 data

Mean of Population 1 data = = 72.125

Mean of Population 2 data = = 73.5

 Hence, = 73.5 - 72.125 = 1.375

= Standard Deviation of paired data = =

                                                                 = 7.3

n = no. of observations = 8

 So, Test Statistics = follows

                         = 0.533

Now, if the critical value of t at 7 degree of freedom is less than the test statistics then we will reject null hypothesis.

At 0.01 level of significance, the t table gives a critical value of 3.499. Since our test statistics is less than the critical value we will not reject null hypothesis and conclude that there is no significant difference between the two population means.