Yes, based on the results of the cross, it appears that these two genes are linked. The results show that the offspring with gray bodies and normal wings, as well as those with black bodies and vestigial wings, are more common than those with a combination of the two traits (gray bodies and vestigial wings, or black bodies and normal wings). This suggests that the genes are located on the same chromosome and are therefore linked.
The number of map units apart that the genes are on the linkage map can be estimated using the ratio of recombination frequency (RF) to the total number of offspring. The RF is the number of offspring with recombinant traits (i.e., gray bodies and vestigial wings, or black bodies and normal wings) divided by the total number of offspring. The map distance is equal to 100% RF divided by the number of offspring with recombinant traits.
In this case, the RF is (50 + 61)/(236 + 253 + 50 + 61) = 111/600 = 0.185, or 18.5%.
So the map distance would be approximately 100% / 18.5% = 5.4 map units. However, this is an estimate and the actual map distance may be slightly different due to various factors such as genetic drift and mutation.
a. Gray body, normal wings