Question

Given two dependent random samples with the following results:

Population 1	53

60

79

78

56

68

60

66

Population 2	57

55

82

68

65

73

51

76

Can it be concluded, from this data, that there is a significant difference between the two population means?

Let d=(Population 1 entry)−(Population 2 entry)

. Use a significance level of α=0.02 for the test. Assume that both populations are normally distributed.

1. State null and alternative hypothesis

2. Find the value of the standard deviation of the paired differences. Round your answer to one decimal place.

3. Compute the value of the test statistic. Round your answer to three decimal places.

4. Determine the decision rule for rejecting the null hypothesis H0. Round the numerical portion of your answer to three decimal places.

5. Make the decision for the hypothesis test.

Answer 1

1. Hypothesis to be tested:
   
   Vs
   
2. Table-1 (Table showing the paired difference)

Population 1	Population 2	Paired Difference(di)
53	57	-4
60	55	5
79	82	-3
78	68	10
56	65	-9
68	73	-5
60	51	9
66	76	-10

Standard Deviation of the paired difference  

3. Test Statistic:

where, s.d : Standard Deviation of the paired difference,    mean of the paired difference
n: sample size

4. Decision Rule:
We will reject the null hypothesis at 0.02 level of significance iff,

where,    is the 0.01 percentile value of the "t-distribution" with 6 degrees of freedom.

i.e.,

5. Decision:

Here,

We fail to reject the null hypothesis at 0.02 level of significance and conclude on the basis of given data that there is no significant difference between the two population means.

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