Problem 2: Two-Sample Inferences A two-sample inference deals with dependent and
ID: 3327360 • Letter: P
Question
Problem 2: Two-Sample Inferences
A two-sample inference deals with dependent and independent inferences. In a two-sample hypothesis testing problem, underlying parameters of two different populations are compared.
In a longitudinal (or follow-up) study, the same group of people is followed over time. Two samples are said to be paired when each data point in the first sample is matched and related to a unique data point in the second sample.
This problem demonstrates inference from two dependent (follow-up) samples using the data from the hypothetical study of new cases of tuberculosis (TB) before and after the vaccination was done in several geographical areas in a country in sub-Saharan Africa. Conclusion about the null hypothesis is to note the difference between samples.
The problem that demonstrates inference from two dependent samples uses hypothetical data from the TB vaccinations and the number of new cases before and after vaccination.
Table 5: Cases of TB in Different Geographical Regions
Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following:
Construct a one-sided 95% confidence interval for the true difference in population means.
Test the null hypothesis that the population means are identical at the 0.05 level of significance.
Geographical regions Before vaccination After vaccination 1 85 11 2 77 5 3 110 14 4 65 12 5 81 10 6 70 7 7 74 8 8 84 11 9 90 9 10 95 8Explanation / Answer
Geographical regions
Before vaccination
After vaccination
Di
(Di-D)
(Di-D)^2
1
85
11
-74
0
0.16
2
77
5
-72
2
2.56
3
110
14
-96
-22
501.76
4
65
12
-53
21
424.36
5
81
10
-71
3
6.76
6
70
7
-63
11
112.36
7
74
8
-66
8
57.76
8
84
11
-73
1
0.36
9
90
9
-81
-7
54.76
10
95
8
-87
-13
179.56
-74
1340.40
D=ED/n=-74
S.D=(E(Di-D)2/n-1)=1340.40/9=12.20
tSTAT=D-µ /S.D/n
Where tSTAT has n - 1 d.f
tSTAT=-74-0/12.20/10
=-19.18
tCRIT=+/-2.26
Reject H0 since tSTAT< tCRIT
Confidence interval: -
-74-2.26*(12.20/10)
-82.72
In minitab, we can calculate the t-test paired data confidence interval using the below procedure.
1)
Enter the data in pre and post columns for before and after vaccination respectively.
2)
Click Stat > Basic Statistics > Paired t. It would amongst the options at the top.
3)
Samples in Columns, First Sample and Second Sample, enter the data respectively and click ok.
4)
The results would be displayed along with the confidence intervals.
Geographical regions
Before vaccination
After vaccination
Di
(Di-D)
(Di-D)^2
1
85
11
-74
0
0.16
2
77
5
-72
2
2.56
3
110
14
-96
-22
501.76
4
65
12
-53
21
424.36
5
81
10
-71
3
6.76
6
70
7
-63
11
112.36
7
74
8
-66
8
57.76
8
84
11
-73
1
0.36
9
90
9
-81
-7
54.76
10
95
8
-87
-13
179.56
-74
1340.40
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