please show all work stairs t taking an elevator. Assume all samples are randoml

Question

please show all work

stairs t taking an elevator. Assume all samples are randomly selected what ls lhe probability that at least 30% ofthe adults in a unple of Isa pre ter tales? i The 2-153) ,0630 b. What is the prohability that between 25 and 40 of the adalts in a sample of 85 nl pfefer to use the stairs? c. Find the probability more than 90 adalts in a sample of 125 prefer the ELEVATOR (ie - do NOT prefer the stairs) d. Find the probability that less than 45% in a sample of 230 adults prefer stairs. nl 178 81

Explanation / Answer

p = 0.36

a)
n = 150
SE = sqrt(p * (1-p)/n) = sqrt(0.36 * (1-0.36)/150) = 0.0392

P(pcap >= 0.3)
= P(z >= (0.3 - 0.36)/0.0392)
= P(z >= -1.5306)
= 0.9371

b)
n = 85
mean = 0.36*85 = 30.6
SE = sqrt(p * (1-p)*n) = sqrt(0.36 * (1-0.36)*85) = 4.4254

P(25 < xbar < 40)
= P((25 - 30.6)/4.4254 < z < (40 - 30.6)/4.4254)
= P(-1.2654 < z < 2.1241)
= 0.8803

c)
here p = 1 - 0.36 = 0.64
n = 125
mean = 0.64*125 = 80
SE = sqrt(p * (1-p)*n) = sqrt(0.36 * (1-0.36)*125) = 5.3666

P(xbar > 90)
= P(z > (90 - 80)/5.3666)
= P(z > 1.8634)
= 0.0312

d)
n = 230
mean = 0.36*230 = 82.8
SE = sqrt(p * (1-p)*n) = sqrt(0.36 * (1-0.36)*230) = 7.2796

xbar = 0.45*230 = 103.5

P(xbar < 103.5)
= P(z > (103.5 - 82.2)/7.2796)
= P(z > 2.926)
= 0.9983

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