Question

7. A recent medical survey reported that 45% of the respondents felt that the doctor explained their condition in a sufficient manner. Assuming that this poll reflects all the patients, find the probability that for 12 patients:

a. Four or more agree
b. No more than 8 agreed

Answer 1

(a) 0.8655

(b)0.8655

Explanation:

X is the number of respodents who agree

P(x ≥4), n=12, p=0.45

P(x ≥4)=1-P(x<4)

P(x<4)=P(0 ≤x<4)= P(0 ≤x≤3)=P(x=0,1,2,3)=P(x=0)+P(x=1)+P(x=2)+P(x=3)

`P(x=0)=12C0(0.45)^(0)(1-0.45)^(12-0)\approx 0.0007662`

`P(x=1)=12C1(0.45)^(1)(1-0.45)^(12-1)\approx 0.007523`

`P(x=2)=12C2(0.45)^(2)(1-0.45)^(12-2)\approx 0.033853`

`P(x=3)=12C3(0.45)^(0)(1-0.45)^(12-3)\approx 0.092326`

`P(x<4)= 0.0007662+0.007523+0.033853+0.092326=0.134468`

P(x ≥4)=1-P(x<4)=1-0.134468=0.8655319

(b)

Y represent those who agree

P(Y≤8), n=12, p=0.45

P(Y≤8)=1-P(Y>8)

P(Y>8)=P(8<Y≤12)=P(9≤Y≤12)=P(Y=9,10,11,12)

`P(Y=9)= 12C9(0.45)^(9)(1-0.45)^(12-9)\approx 0.092326`

`P(Y=10)= 12C10(0.45)^(10)(1-0.45)^(12-10)\approx 0.033853`

`P(Y=11)= 12C11(0.45)^(11)(1-0.45)^(12-11)\approx 0.007523`

`P(Y=12)= 12C12(0.45)^(12)(1-0.45)^(12-12)\approx 0.0007662`

Total =0.134468

P=1-0.134468=0.8655319