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C4, L2, S1 Probabilities and Proportions Probabilities and proportions are numerically equivalent. (i.e. they convey the same information.) e.g. The proportion.

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Presentation on theme: "C4, L2, S1 Probabilities and Proportions Probabilities and proportions are numerically equivalent. (i.e. they convey the same information.) e.g. The proportion."— Presentation transcript:

1 C4, L2, S1 Probabilities and Proportions Probabilities and proportions are numerically equivalent. (i.e. they convey the same information.) e.g. The proportion of U.S. citizens who are left handed is 0.1; a randomly selected U.S. citizen is left handed with a probability of approximately 0.1.

2 C4, L2, S2 An experiment was conducted to study the effect of questionnaire layout and page size on response rate in a mail survey. 3. Questionnaire Format Example 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat

3 C4, L2, S3 3. Questionnaire Format Example LetA be the event that the questionnaire was Typeset, Large Page. B be the event that the questionnaire was Responded to. C be the event that the questionnaire was Typewritten, Large Page. 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat

4 C4, L2, S4 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat (a)What proportion of the sample responded to the questionnaire? A: Typeset, Large B: Response C: Typewritten, Large P(B) = 3. Questionnaire Format Example

5 C4, L2, S5 (a)What proportion of the sample responded to the questionnaire? P(B) = 541/856 = 0.632 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large 3. Questionnaire Format Example

6 C4, L2, S6 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat b)What proportion of those who received a typeset large page questionnaire actually responded to the questionnaire? A: Typeset, Large B: Response C: Typewritten, Large 3. Questionnaire Format Example

7 C4, L2, S7 FormatResponses Non- responses Total Typewritten, Small Page8657143 Typewritten, Large Page19169270 Typeset,Small Page7269141 Typeset,Large Page19292284 Total541315856 b)What proportion of those who received a typeset large page questionnaire actually responded to the questionnaire? 3. Questionnaire Format Example A: Typeset, Large B: Response C: Typewritten, Large P(B|A) =

8 C4, L2, S8 b)What proportion of those who received a typeset large page questionnaire actually responded to the questionnaire? P(B|A) = 192/284 = 0.676 3. Questionnaire Format Example 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large

9 C4, L2, S9 c)What proportion of the sample received a typeset large page questionnaire and responded? 3. Questionnaire Format Example 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat P(A and B) = A: Typeset, Large B: Response C: Typewritten, Large

10 C4, L2, S10 c)What proportion of the sample received a typeset large page questionnaire and responded? P(A and B) = 192/856 = 0.224 3. Questionnaire Format Example 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large

11 C4, L2, S11 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat d)What proportion of the sample responded to a large page questionnaire (either Typewritten or Typeset)? 3. Questionnaire Format Example A: Typeset, Large B: Response C: Typewritten, Large P(B and (A or C ) )

12 C4, L2, S12 d)What proportion of the sample responded to a large page questionnaire (either Typewritten or Typeset)? P(B and (A or C ) ) = (192+191)/856 = 383/856= 0.447 3. Questionnaire Format Example 856315541Total 28492192Typeset,Large Page 1416972Typeset,Small Page 27069191Typewritten, Large Page 1435786Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large

13 C4, L2, S13 2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991) Brain Injury No Brain Injury Row Totals No Helmet 9719182015 Helmet Worn 17977994 Column Totals11428953009 BI = the event the motorcyclist sustains brain injury NBI = no brain injury H = the event the motorcyclist was wearing a helmet NH = no helmet worn P(BI) = 114 / 3009 =.0379 What is the probability that a motorcyclist involved in a accident sustains brain injury?

14 C4, L2, S14 2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991) Brain Injury No Brain Injury Row Totals No Helmet 9719182015 Helmet Worn 17977994 Column Totals1142893009 BI = the event the motorcyclist sustains brain injury NBI = no brain injury H = the event the motorcyclist was wearing a helmet NH = no helmet worn P(H) = 994 / 3009 =.3303 What is the probability that a motorcyclist involved in a accident was wearing a helmet?

15 C4, L2, S15 2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991) Brain Injury No Brain Injury Row Totals No Helmet 9719182015 Helmet Worn 17977994 Column Totals11428953009 What is the probability that the cyclist sustained brain injury given they were wearing a helmet? P(BI|H) = 17 / 994 =.0171 BI = the event the motorcyclist sustains brain injury NBI = no brain injury H = the event the motorcyclist was wearing a helmet NH = no helmet worn

16 C4, L2, S16 2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991) Brain Injury No Brain Injury Row Totals No Helmet 9719182015 Helmet Worn 17977994 Column Totals11428953009 What is the probability that the cyclist not wearing a helmet sustained brain injury? P(BI|NH) = 97 / 2015 =.0481 BI = the event the motorcyclist sustains brain injury NBI = no brain injury H = the event the motorcyclist was wearing a helmet NH = no helmet worn

17 C4, L2, S17 2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991) Brain Injury No Brain Injury Row Totals No Helmet 9719182015 Helmet Worn 17977994 Column Totals11428953009 How many times more likely is a non-helmet wearer to sustain brain injury?.0481 /.0171 = 2.81 times more likely. This is called the relative risk or risk ratio (denoted RR).

18 C4, L2, S18 Building a Contingency Table from a Story 3. HIV Example A European study on the transmission of the HIV virus involved 470 heterosexual couples. Originally only one of the partners in each couple was infected with the virus. There were 293 couples that always used condoms. From this group, 3 of the non-infected partners became infected with the virus. Of the 177 couples who did not always use a condom, 20 of the non- infected partners became infected with the virus.

19 C4, L2, S19 Let C be the event that the couple always used condoms. (NC be the complement) Let I be the event that the non-infected partner became infected. (NI be the complement) CNC NI I 3. HIV Example Total Condom Usage Infection Status

20 C4, L2, S20 A European study on the transmission of the HIV virus involved 470 heterosexual couples. Originally only one of the partners in each couple was infected with the virus. There were 293 couples that always used condoms. From this group, 3 of the non-infected partners became infected with the virus. CNC NI I 3. HIV Example Total Condom Usage Infection Status 470 293 3

21 C4, L2, S21 Of the 177 couples who did not always use a condom, 20 of the non-infected partners became infected with the virus. CNC NI I 3. HIV Example Total Condom Usage Infection Status 470 293 320 177 290157 23 447

22 C4, L2, S22 a)What proportion of the couples in this study always used condoms? CNC NI I Total Condom Usage Infection Status 470 293 320 177 290157 23 447 3. HIV Example P(C )

23 C4, L2, S23 a)What proportion of the couples in this study always used condoms? CC I I Total Condom Usage Infection Status 470 293 320 177 290157 23 447 3. HIV Example P(C ) = 293/470 (= 0.623)

24 C4, L2, S24 b)If a non-infected partner became infected, what is the probability that he/she was one of a couple that always used condoms? 3. HIV Example CNC NI I Total Condom Usage Infection Status 470 293 320 177 290157 23 447 P(C|I ) = 3/23 = 0.130

25 C4, L2, S25 3. Death Sentence Example University of Florida sociologist, Michael Radelet, believed that if you killed a white person in Florida the chances of getting the death penalty were three times greater than if you had killed a black person. In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death sentence had murdered a white person. (Gainesville Sun, Oct 20 1986)

26 C4, L2, S26 3. Death Sentence Example Let W be the event that the victim is white. B be the event that the victim is black. D be the event that the sentence is death. ND be the event that the sentence is not death. ND D W Victim’s Race Sentence Total B

27 C4, L2, S27 3. Death Sentence Example In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death sentence had murdered a white person. ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184

28 C4, L2, S28 3. Death Sentence Example a)What proportion of the murderers in this study received the death sentence? P(D) = ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184

29 C4, L2, S29 3. Death Sentence Example a)What proportion of the murderers in this study received the death sentence? ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184 P(D) = 36/326 = 0.1104

30 C4, L2, S30 3. Death Sentence Example b)If a victim from this study was white, what is the probability that the murderer of this victim received the death sentence? ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184 P(D|W ) =

31 C4, L2, S31 3. Death Sentence Example b)If a victim from this study was white, what is the probability that the murderer of this victim received the death sentence? ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184 P(D|W ) =

32 C4, L2, S32 3. Death Sentence Example b)If a victim from this study was white, what is the probability that the murderer of this victim received the death sentence? ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184 P(D|W ) = 30/214 = 0.1402

33 C4, L2, S33 3. Death Sentence Example c)If a victim from this study was black, what is the probability that the murderer of this victim received the death sentence? P(D|B ) = ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184

34 C4, L2, S34 3. Death Sentence Example c)If a victim from this study was black, what is the probability that the murderer of this victim received the death sentence? P(D|B ) = ND D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184

35 C4, L2, S35 3. Death Sentence Example c)If a victim from this study was black, what is the probability that the murderer of this victim received the death sentence? P(D|B ) = 6/112 = 0.0536 P(D|W) is approx. three times larger than P(D|B) D D W Victim’s Race Sentence Total B 326 290 36 30 6 112214 106184

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