STAB22H3 Lecture Notes - Dont, Gardasil
CHAPTER 22 - COMPARING TWO PROPORTIONS
WHERE ARE WE GOING?
- comparing 2 prop's
- want to see whether 2 groups are diff, or do they vary by chance
main text, p585
(EX- MASSA Drivers)
- 6971 male drivers
- seatbelt use
- highway safety
- n = 161 loc's in Massachussets, using SRS
- F drivers wore belt >70% of the time, regardless of gender of passenger(s)
- out of 4,208 M drivers w/ F passengers, 2777 (66%) wore belts
- out of 2,763 M drivers w/ M passengers, only 1363 (49.3%) wore belts
WHY COMPARE BETWEEN TWO PROPORTIONS (aka PERCENTAGES)?
- interested in finding out how 2 groups differ
(ex) is exptl treatment better than placebo?
(EX- MASSA Drivers)
- know: diff. in prop's of men wearing sealtbelts from sample
- 66% - 49.3% = 16.7%
- more interested in: true difference for ALL men?
- it is not likely that the diff. we obtained is the truth, b/c prop's will vary from sample to sample
- to do this, req. a new ruler: SD of samplign distribution model for diff. in prop's
The variance of sum or diff. of 2 indep. random var's is sum of their variances
=> aka for indep. random var's, variances always add (regardless of whether you are adding or
subtracting the 2 random var's)
WHY DOES VARIATION INCREASE, DESPITE SUBTRACTING TWO RANDOM
(An.) - bowl of cereal
- cereal box claims that there is 16oz of cereal in it
- this is not exact: b/c there is small variation from box to box
- when portion of cereal is poured into bowl (we want 2oz serving), we know that it will not be
exact; there is variation assoc. w/ this too
- qn: how much cereal is left in the box?
- is the guess more closer to guess of full box?
- AFTER cereal is poured into bowl, amt of cereal in box still remains a random quantity (but
smaller mean now), BUT it is even more variable b/c of additional variation in amt that was
- variance in amt of cereal remaining in box = sum of 2 variances
- becomes more variable, now that it has been distributed into two containers
- this formula for SD ONLY works for INDEPENDENT RANDOM VARIABLES
=> must check for independence b4 using it