Well... not sure now much help this is. The pattern doesn't give a gauge! At least not that I found. The directions also don't match up with the number of stitches. I THINK I know what they intended, but it's not what they wrote. For example, in this line : next row (RS): K5, sl1, K1, psso, K4 K5 You are starting with 42 stitches. I suspect they want the sl1, K1, psso, K4 repeated 5 times, but that still leaves you with the wrong number of stitches, so I suspect you don't K4 the 5th time but leave off the K4 and do that last K5 instead.
You may want to consider a different pattern if you don't want to do some refiguring and guessing.
If this is the hat that you are wanting to knit regarldess...
Andean Trail by Cherry Tree Hill, the yarn they are using, is listed as averaging 2 stitches per inch on size 11 needles. Not sure what it would get on the size 15 needles the band is done on, and of course this changes with every knitter!
Do you know the size of the head of the child you are knitting for? Basically I would cast on enough for 1" less than the child's head size.
If you don't know the child's head size, I would probably knit the hat to come about 18" around the band. If the child has a head on the smaller end, it might be rather loose, but should stretch fine for up to a 20" head. That would fit the majority of children in that age range.
Do you already have a yarn in mind? That would affect my choice of needle size. You can either change the number of stitches cast on to match the size head you want (and make corresponding changes to the decreases) or you can use a yarn that works with smaller needles.
Since this yarn is listed on size 11 needles, you might be able to use it on size 8's with 11's for the Brim, but I wouldn't know how tight it knit up and how it looked without trying it.
The other thing you need to do it make sure that the height of the hat is sufficient for the child it's being knit for.
I have probably confused you more than helped, but I'll post this anyway just in case it helps. I'm sure Ingrid could put all this much more succienctly.