Question

to the previous experience, only 20% of the elderly will welcome the visit while 80% of the elderly will reject the visit. Mary, one of the part-time interviewers, will be responsible to visit 13 elderly residents today. Use X to denote the number of successful visits she will make today.
(b) Most likely, how many elderly residents will welcome Mary’s visit? Calculate the corresponding probability.
(c) Suppose Mary will spend 5 minutes in each rejected visit and 30 minutes in each successful visit. Use Y to denote the total time Mary will spend today. Express Y in terms of X. Hence, find the (i)minimum, (ii) maximum, and (iii) expected total time (in minutes) for Mary to finish all visits today?
(d) Mary will get $30 allowance for each rejected visit and $100 allowance for each successful visit. UseW to denote the total allowance Mary will get today. Express W in terms of X. Hence, find the (i)minimum, (ii) maximum, and (iii) expected total allowance Mary will get today?

Answer 1

(b) $E(X)=pn=0.2*13=2.6$

The most likely outcome satisfies following inequality.

$np-q \leq k \leq np+p$

$13 \cdot 0.2-0.8 \leq k \leq 13 \cdot 0.2+0.2$

$1.8 \leq k \leq 2.8$

Thus, most likely, 2 elderly residents will welcome Mary’s visit.

The corresponding probability: $P(X=2)=C_{13}^2(0.2)^2(1-0.2)^{11}=0.2680$

(c) $Y=30X+5(13-X)=25X+65$

(i) $Y_{min}=25*0+65=65\;min$

(ii) $Y_{max}=25*13+65=390\;min$

(iii) $Y_{exp}=25*2.6+65=130\;min$

(d) $W=100x+30(13-x)=70X+390$

(i) $W_{min}=70*0+390=390$

(ii) $W_{max}=70*13+390=1300$

(iii) $W_{exp}=70*2.6+390=572$