To compute the test statistic for comparing two matched pairs in hypothesis testing, we follow several steps:
1. Determine the differences between each pair:
Given pairs from the samples are:
- [tex]\( (18, 19) \)[/tex]
- [tex]\( (11, 11) \)[/tex]
- [tex]\( (19, 19) \)[/tex]
- [tex]\( (20, 20) \)[/tex]
- [tex]\( (21, 16) \)[/tex]
The differences,
[tex]\[
d_i = \text{Sample 1} - \text{Sample 2}
\][/tex]
are as follows:
[tex]\[
d = [18-19, 11-11, 19-19, 20-20, 21-16] = [-1, 0, 0, 0, 5]
\][/tex]
2. Calculate the mean of the differences:
The mean difference, [tex]\( \bar{d} \)[/tex], is calculated by summing up the differences and dividing by the number of differences [tex]\( n \)[/tex]:
[tex]\[
\bar{d} = \frac{-1 + 0 + 0 + 0 + 5}{5} = \frac{4}{5} = 0.8
\][/tex]
3. Calculate the standard deviation of the differences:
The standard deviation of the differences, [tex]\( s_d \)[/tex], is:
[tex]\[
s_d = 2.3874672772626644
\][/tex]
4. Calculate the test statistic:
The test statistic for a paired t-test is calculated using:
[tex]\[
t = \frac{\bar{d}}{s_d / \sqrt{n}}
\][/tex]
where [tex]\( n = 5 \)[/tex] is the number of pairs.
Substitute the values:
[tex]\[
t = \frac{0.8}{2.3874672772626644 / \sqrt{5}} = 0.7492686492653552
\][/tex]
Given the choices, the correct test statistic is:
C) 0.749
