The capacity of an elevator is 8 people or 1232 pounds. the capacity will exceed

Question

The capacity of an elevator is 8 people or 1232 pounds. the capacity will exceed if 8 people have weights with a mean greater than 1232/8= 15 pounds. Suppose the people have weights that are normally distributed with a mean of 163 pounds and a standard deviation of 30 pounds.

a) Find the probability that if a person is randomly selected, his weight will be greater than 154 pounds.

b) Find the probability that 8 randomly selected people will have a mean that's greater than 154 pounds.

c) Does the elevator appear to have the correct weight limit?

Explanation / Answer

At the very outset, the second line in the question says '1232/8= 15 pounds' which is taken as a typo and the solution is based on 1232/8 = 154

Solution

Back-up Theory

If a random variable X ~ N(µ, 2), i.e., X has Normal Distribution with mean µ and variance 2, then

Z = (X - µ)/ ~ N(0, 1), i.e., Standard Normal Distribution ………………………..(1)

P(X or t) = P[{(X - µ)/ } or {(t - µ)/ }] = P[Z or {(t - µ)/ }] .………(2)

X bar ~ N(µ, 2/n),…………………………………………………………….…….(3),

where X bar is average of a sample of size n from population of X.

So, P(X bar or t) = P[Z or {(n)(t - µ)/ }] …………………………………(4)

Probability values for the Standard Normal Variable, Z, can be directly read off from

Standard Normal Tables or using Excel Function……………………………………..(5)

Now, to work out solution,

Let X = weight of people. Then, given,

‘Suppose the people have weights that are normally distributed with a mean of 163 pounds and a standard deviation of 30 pounds.’

X ~ N(163, 302)

Part (a)

Probability that a person randomly selected, will have weight greater than 154 pounds = P(X > 154)

= P[Z > {(154 - 163)/30}] = P(Z > - 0.3) = 1 - P(Z - 0.3)

= 1 – 0.3821 = 0.6179 ANSWER

[vide (2) under Back-up Theory]

Part (b)

Probability that 8 randomly selected people will have a mean that's greater than 154 pounds = P(Xbar > 154)

= P[Z > {(8)(154 - 163)/30}] [vide (4) under Back-up Theory]

= P(Z > - 0.8485) = 1 - P(Z - 0.8485)

= 1 – 0.1981 = 0.8019 ANSWER

[vide (2) under Back-up Theory]

Part (c)

Does the elevator appear to have the correct weight limit?

Clearly, weight limit is unrealistic since given height distribution of the population,

80% of the time, people cannot use the elevator. ANSWER

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