To determine the number of individuals in an independent-measures experiment, we need to consider the degrees of freedom (df) associated with the t-test reported by the researcher. The degrees of freedom for an independent-measures t-test, which involves two sample groups, is calculated using the formula:
[tex]\[ df = N_1 + N_2 - 2 \][/tex]
where [tex]\( N_1 \)[/tex] and [tex]\( N_2 \)[/tex] are the sample sizes of the two groups. In this case, the researcher reports [tex]\( t(9) \)[/tex], indicating that the degrees of freedom (df) is 9.
Assuming equal sample sizes for the two groups (i.e., [tex]\( N_1 = N_2 = N \)[/tex]), we can set up the equation as follows:
[tex]\[ df = N + N - 2 \][/tex]
Substitute the given degrees of freedom into the equation:
[tex]\[ 9 = 2N - 2 \][/tex]
To solve for [tex]\( N \)[/tex], follow these steps:
1. Add 2 to both sides of the equation to isolate the term involving [tex]\( N \)[/tex]:
[tex]\[ 9 + 2 = 2N \][/tex]
2. Simplify the equation:
[tex]\[ 11 = 2N \][/tex]
3. Divide both sides by 2 to solve for [tex]\( N \)[/tex]:
[tex]\[ N = \frac{11}{2} \][/tex]
[tex]\[ N = 5.5 \][/tex]
Since [tex]\( N \)[/tex] must be a whole number (as you can't have a fraction of an individual in a sample), it implies that each group has 5 individuals. Therefore, the total number of individuals in the study is:
[tex]\[ N_1 + N_2 = 5 + 5 = 10 \][/tex]
So, there are 10 individuals in this study.
