You observe a population of Island Sparrows for which the phenotype of crown color varies from yellow to white. You perform two experiments. Experiment 1 shows that this variation in the phenotype affects mating success, so that birds with yellow crown colors have more offspring. Experiment 2 shows that this variation in phenotype is determined fully by the diet of the birds and not genetics. Can crown color evolve via natural selection in this population of Island Sparrows?

Yes, the color of the crown evolves by the process of natural selection.

Explanation:

Based on experiment 1 mentioned in the question, it is clear that the sparrows with the yellow crown are demonstrating reproductive success, and based on experiment 2, it is demonstrated that the origination of crown color is solely dependent upon the diet consumed by the sparrows, not on the genes.  

The yellow crown sparrows exhibit enhanced skills in locating food, finding food or prey is dependent upon the factors like skills of hunting, vision, the musculature of the body, and other things, and all these are partially or completely determined by genetics. This yellow crown sparrow will further pass on their genes to their offspring, which will further provide them good skills of locating food, and thus, will obtain the diet needed for proper growth. These offspring will further pass on the traits to the future generations, and so on.  

On the other hand, the sparrows exhibiting white crown seem to be less efficient in locating their food, and they are also generating fewer offspring that exhibit the same lack of food locating skills like their parents. Due to less tendency of finding food, the sparrows with the white crown will suffer from poor diet and thus, will further generate less offspring, and so on.  

Therefore, it is clear that for betterment, the sparrows with the yellow crown will evolve further due to their better skills of locating food, and this gives rise to the phenomenon of natural selection.  

i think that is new experiments yield untrustworthy data and should be carefully considered, because you don't know what will happen with the experiments.

the answer is sex-linked recessive.