(a)
(i) We can combine the means and variances using simple arithmetic: combined means=6+12=18, combined variances=6+9=15. So combining the datasets (we end up with a dataset, the size of which is the product of the sizes of the datasets) we can work out Z values=(X+W-18)/√15. We need the standard deviation of the combined set so we take the square root of the combined variance.
(ii) Combined means 6-12=-6, combined variances=6+9=15, because variance is always positive.
(b)
For X+W=12.87, Z=(12.87-18)/√15=-1.32456 approx.
For X+W=18.32, Z=(18.32-18)/√15=0.08262 approx.
From normal distribution tables, the probability for Z=-1.32 is 1-p(1.32)=1-0.9066=0.0934 and 0.5319 for Z=0.08. The probability of being within these limits is 0.5319-0.0934=0.4385 or 43.85%.