I got the information from the PDF that you linked in your first post.  It says that Word (byte, really) 16 is the checksum, and Word 16 is chosen to make words 8 through 16 add up to 0x100, or 0x200, etc.  Put another way, Word 16 is chosen to make the last two digits of the sum be 00.  So to compute Word 16, add up words 8-15, and subtract the least 2 digits from 0x100.

In IR signals, we can send the least significant bit first or send the most significant bit first.  It's just a choice that the designers of the airconditioner made. All the data in the PDF is arranged so that each word is to be sent with the least significant bit first.  But IRScope presents the bits in the order received, and shows the first bit received on the left side of the word, while the PDF is setup so that the bit which is received first is put on the right side of the word.  So we have to reverse the bits provided by IRScope to get the LSB form that matches the PDF.  If we don't do that, the checksum formula won't work.

Once the check sum is calculated, it needs to be reversed, because Pronto Hex needs the bits in the order that they will be transmitted.

Now recall that the decode from IRScope gave just 5 bits of Word 16: 11010, meaning that we don't know the last 3 bits--Word 16 is 11010???
Our reversed Word 16 that we just calculated is 11010010.  Our calculated Word 16 agrees with the decode for the first 5 digits (11010) and we then insert the last 3 bits into the Pronto Hex.