ANSWERS TO ACTIVITIES

TOPIC 16 - Learning About a Proportion

Activity 16-1

(a) .75

(b) .5

(c)Ý Table after observing 1^st data result = "acceptable" should look like

MODEL	PRIOR	LIKE	PRODUCT	POST
p=.75	.9	.75	.675	.931
p=.5	.1	.5	.050	.069
SUM	Ý		.725	1.000

(g) Completed table after observing 2^nd data result = "defective":

MODEL	PRIOR	LIKE	PRODUCT	POST
p=.75	.931	.25	.233	.869
p=.5	.069	.5	.035	.131
SUM	Ý		.268	1.000

(h) Completed table after observing 3^rd data result = "defective":

MODEL	PRIOR	LIKE	PRODUCT	POST
p=.75	.869	.25	.217	.770
p=.5	.131	.5	.065	.230
SUM	Ý		.282	1.000

(g) Completed table after observing 4^th data result = "defective":

MODEL	PRIOR	LIKE	PRODUCT	POST
p=.75	.770	.25	.192	.627
p=.5	.230	.5	.115	.373
SUM	Ý		.307	1.000

(j) Likelihood for p=.75: LIKE = .75 x .25 x .25 x .25 = .0117

ÝÝÝÝ Likelihood for p = .25: LIKE = .5 x .5 x .5 x .5 = .0625

MODEL	PRIOR	LIKE	PRODUCT	POST
p=.75	.9	.0117	.105	.629
p=.5	.1	.0625	.062	.371
SUM	Ý		.167	1.000

Activity 16-2

(a) Likelihood of (H,H,O,O) for p=.3 is LIKE = .3 x .3 x .7 x .7 = .0441

ÝÝÝÝÝÝÝ Likelihood of (H,H,O,O) for p = .4 is LIKE = .4 x .4 x .6 x .6 = .0576

(b)ÝÝÝÝÝÝÝÝÝÝ

ÝMODEL	PRIOR	LIKE	PRODUCT	POST
p = .2	.4	.0256	.0102	.278
p = .3	.59	.0441	.0260	.706
p = .4	.01	.0576	.0006	.016
SUM			.0368	1.000

Activity 16-3

Let's suppose I obtained 2 heads in 5 spins -- I have the following posterior probabilities:

ÝÝÝÝÝ pÝÝÝÝÝÝÝ	post
ÝÝÝÝÝÝÝ 0 ÝÝÝÝ	0
ÝÝÝ 0.1000	0.0440
ÝÝÝ 0.2000	0.1230
ÝÝÝ 0.3000	0.1850
ÝÝÝ 0.4000	0.2070
ÝÝÝ 0.5000	0.1880
ÝÝÝ 0.6000	0.1380
ÝÝÝ 0.7000	0.0790
ÝÝÝ 0.8000	0.0310
ÝÝÝ 0.9000	0.0050
0	post

(b) The most likely value of p is .4 -- this value has a probability of .207.

(c) ÝThe probability that p is equal to .7 is .079

(d) The probability that p is greater than 5

= prob (p = .6, .7, .8, .9, 1)

= .138 + .079 + .031 + .005 + 0 = .253

(e) ordering values of p by most likely to least likely

ÝÝÝÝÝ pÝÝÝÝÝÝÝ	post	cum prob
.4	0.2070	0.2070
.5	0.1880	0.3950
.3	0.1850	0.5800
.6	0.1380	0.7180
.2	0.1230	0.8410
.7	0.0790	0.9200
.1	0.0440	0.9640
.8	0.0310	0.9950
.9	0.0050	1.0000
1	0	1.0000
0	0	1.0000

(f) 50% probability set is {.3, .4, .5}

90% probability set is {.2, .3, .4, .5, .6, .7}

Activity 16-4

(c) 16/22 = .727

(d) No, since you are only considering couples with different ages.

(e) The most likely values are p = .71, .72, .73, .74

(f) Prob(p is larger than .5) = 1 - Prob(p is .5 or smaller)

= 1 - Prob(p is 0, .01, .02, Ö, .5)

= 1 - [.001 + .001 + .001 + .001 + .002 + .002 + .003 + .003 + .004]

= 1 - .018 = .982 .

(g) I put in the 15 most likely values of p and the associated probabilities in the table below:

PROPORTION p	PROBABILITY
.71, .72, .73, .74	.043, .043, .043, .043
.70, .75Ý	.042, .042
.76, .69ÝÝÝÝ	.041, .040
.68, .77, .67, .78	.039, .039, .037, .037
.66, .79, .65ÝÝ	.034, .034, .032

ÝÝÝÝÝÝÝÝÝÝÝÝ The total probability on the right is .043 + .043 + Ö+ .032 = .589

ÝÝÝÝÝÝÝÝÝÝÝ So the set of values {.71, .72, Ö, .65} has total probability .589

(h) The proportion p lies between .65 and .79 with probability .589 - that is, [.65, .79] is a 58.9% probability interval.

ÝÝÝÝ Activity 16-5

(a) IfÝ I am very skeptical that Frank has ESP, I would assign the model p = .5 a small probability.

My prior might be

MODEL	PRIOR
p = .25ÝÝÝÝÝÝÝÝ	.9ÝÝÝÝÝÝÝÝ
p = .5ÝÝÝÝÝÝÝÝÝÝ	.1ÝÝÝÝÝÝÝÝ

Your prior could be different, but the model p = .5 should be given a small probability.

(c) data is C = Frank got card correct

MODEL	PRIOR	ÝÝÝ LIKEÝÝÝ	PRODUCT	POSTERIOR
p = .25ÝÝÝÝÝÝÝÝ	.9ÝÝÝÝÝÝÝÝ	.25ÝÝÝÝÝÝÝÝÝ	.225	.818
p = .5ÝÝÝÝÝÝÝÝÝÝ	.1ÝÝÝÝÝÝÝÝ	.5ÝÝÝÝÝÝÝÝÝÝÝ	.050ÝÝÝÝÝÝÝÝÝÝÝÝÝ	.182
SUM			.275	1.000

(d) Frank got second card wrong (W)

MODEL	PRIOR	ÝÝÝ LIKEÝÝÝ	PRODUCT	POSTERIOR
p = .25ÝÝÝÝÝÝÝÝ	.818ÝÝÝÝÝÝÝÝ	.75ÝÝÝÝÝÝÝÝÝ	.614	.871
p = .5ÝÝÝÝÝÝÝÝÝÝ	.182ÝÝÝÝÝÝÝÝ	.5ÝÝÝÝÝÝÝÝÝÝÝ	.091	.129
SUM			.705	1.000

ÝÝÝ Activity 16-6

(a)

MODEL	PRIOR
p=.5	1/3 = .33
p=.67	1/3 = .33
p=.83	1/3 = .33

(b) Data is G, G, G, M

ÝÝ LIKELIHOODS:

ÝÝÝÝÝÝÝÝ if p = .5, prob(G, G, G, M) = .5 x .5 x .5 x .5 = .0625

ÝÝÝÝÝÝÝÝ if p = .67, prob(G, G, G, M) = .67 x .67 x .67 x .33 = .0993

ÝÝÝÝÝÝÝÝ if p = .83, prob(G, G, G, M) = .83 x .83 x .83 x .17 =Ý .0972

(d)

MODEL	PRIOR	LIKE	PRODUCT	POSTERIOR
p=.5	1/3 = .33	.0625ÝÝÝÝÝÝÝ	.0206ÝÝÝÝÝÝÝÝÝÝ	.241
p=.67	1/3 = .33	.0993ÝÝÝÝÝÝÝ	.0328ÝÝÝÝÝÝÝÝÝÝ	.383
p=.83	1/3 = .33	.0972ÝÝÝÝÝÝÝ	.0321ÝÝÝÝÝÝÝÝÝÝ	.375
SUM			.0855

Ý Activity 16-8

(d) Of the 20 pages, 6 contain ads, so I guess p is in the neighborhood of 6/20 = .3

(e)ÝÝ The 3 most likely values of p are .2, .3, .4

(f)Ý The probability that p is either .2, .3, .4Ý = .229 + .403 + .261 = .893

(g)Ý Prob( p is at least .5) = Prob( p is .5, .6, Ö, 1) = .078 + .010 + 0 + Ö+ 0 = .088.