A Pearson correlation coefficient [tex]\( r = +1.00 \)[/tex] between [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] indicates a perfect positive linear relationship between the two variables. This means that for every unit increase in [tex]\( X \)[/tex], there is a consistent and predictable increase in [tex]\( Y \)[/tex], and the change in [tex]\( Y \)[/tex] can be described exactly by a linear function of [tex]\( X \)[/tex]. Let’s analyze the given choices to understand what a [tex]\( r = +1.00 \)[/tex] implies:
1. Every change in [tex]\( X \)[/tex] causes a change in [tex]\( Y \)[/tex].
- This statement is true because the Pearson correlation of [tex]\( +1.00 \)[/tex] implies that [tex]\( Y \)[/tex] changes perfectly in response to any change in [tex]\( X \)[/tex].
2. Each time [tex]\( X \)[/tex] increases, there is a perfectly predictable increase in [tex]\( Y \)[/tex].
- This statement is also true because with [tex]\( r = +1.00 \)[/tex], the relationship between [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] is perfectly linear and positive, meaning that increases in [tex]\( X \)[/tex] always result in predictable increases in [tex]\( Y \)[/tex].
3. All of the other 3 choices occur with a correlation of [tex]\( +1.00 \)[/tex].
- This statement is a summary that will be true if all the other individual statements are true.
4. Every increase in [tex]\( X \)[/tex] causes an increase in [tex]\( Y \)[/tex].
- This statement is true as well since [tex]\( r = +1.00 \)[/tex] indicates a perfect positive relationship, meaning increases in [tex]\( X \)[/tex] will result in increases in [tex]\( Y \)[/tex].
Since each of the first three statements is true, the final choice, "All of the other 3 choices occur with a correlation of [tex]\( +1.00 \)[/tex]" is also true. Therefore, the correct choice is:
[tex]\[ \boxed{3} \][/tex]
