The null hypothesis for the interaction states that (in the population) the linear relationship between X and Y is the same at all levels of W, your categorical variable. The F-test for the interaction is not statistically significant at the .05 level (p = .0556). Therefore, you do not have sufficient evidence to reject that null hypothesis (assuming you've set alpha = .05).

The t-tests for the simple slopes (i.e., the slopes within each level of W) test a different null hypothesis: The null hypothesis for each of those tests says that in the population, the slope = 0. For levels 1, 2 and 5, the p-values are well above .05, so you cannot reject H0 for those levels. But for levels 3 and 4, p is well below .05, so you can reject H0.

Because the test for interaction and the tests on the simple slopes test different null hypotheses, there is nothing inconsistent in what you are seeing.

HTH.

Also be very aware that a failure to reject the null hypothesis is not sufficient to prove the null hypothesis true. The dichotomy (to the extent that it exists) is "reject the null hypothesis" versus "inconclusive outcome." So the outcome that fails to reject the null hypothesis is not inconsistent with existing theory.

This might help: https://amstat.tandfonline.com/doi/full/10.1080/00031305.2019.1583913#.XJbcU_ZFxPY

Nonsignificance of R squares means that model doesn't contain all the main variables ;but the c

But significant of effect denotes that variable has significant effect but not unique; there are another variables also has significant effects.

Hi Bruce Weaver

Thank you for the excellent response!

I only have one remaining question...

If my results are above .05 in testing the first null hypothesis (and thus I fail to reject that null hypothesis), is there any need to consider and/or report the results of the second hypothesis?

Thank you in advance

Hello Edward. What you're calling the second hypothesis, if I follow, is really a set of hypotheses: For each simple slope, the null hypothesis states that in the population, the simple slope = 0. As to whether you proceed with any or all of those tests, do any of those tests correspond to a priori research questions? If so, I would go ahead with them. If they are addressing questions that arose only after you saw the data, take care to treat the results as exploratory rather than confirmatory--or as some folks put it, as hypothesis-generating rather than hypothesis-confirming. This could include using some kind of correction for multiplicity. HTH.