An elevator is designed based on the assumption that people’s weights are Normal

Question

An elevator is designed based on the assumption that people’s weights are Normally distributed with a mean of 100 kg and a standard deviation of 10 kg.

(a) According to this model, what is the probability that a random person weighs in the range [80,90]?

(b) Assume 4 people are in the elevator. What is the probability that the total weight is more than 440kg?

(c) Assume 4 people are in the elevator. What is the probability that the sample mean of the weights for these 4 people is greater than 110kg?

The elevator is designed to sound an alarm if the total weight in the elevator is greater than 950 kg. A sign in the elevator reads: "Max Persons: N". You desire to find the maximal integer N while ensuring that the chance of sounding an alarm at full capacity is less than 1 percent. What is N ?

Explanation / Answer

Mean is 100 and SD is 10

a) P(80<x<90)=P((80-100)/10<Z<(90-100)/10)=P(-2<Z<-1) which is also equal to P(1<Z<2) . This is P(Z<2)-P(Z<1). From the normal distribution tables, we get 0.9772-0.8413=0.1359

b) We can look at weight of one person thus it is 440/4=110. we need to find P(x>110)=P(Z>(110-100)/10)=P(Z>1)=1-P(Z<1)=1-0.8413=0.1587

c) here we need to find probability that the sample mean of the weights for these 4 people is greater than 110kg

therefore P(xbar>110)=P(Z>(110-100)/(10/sqrt(4)))=P(Z>2)=1-P(Z<2)=1-0.9772=0.0228

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