Mathematics, 05.05.2020 17:01 raven681

The capacity of n elevator is 12 people or 1968 pounds. The capacity will be exceeded if 12 people have weights with a mean greater than 1968/12=164 pounds. Suppose the people have weights that are normally distributed with a mean of 171 lb and a standard deviation of 34 lb.

a. find the probability that if a person is randomly selected, his weight will be greater than 164 pounds.

The probability is approximately ___

b. Find the probability that 12 randomly selected people will have a neam that is greater than 164 pounds.

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Answer from: GodlyGamer8239

a) 58.32% probability that his weight will be greater than 164 pounds.

b) 76.11% probability that 12 randomly selected people will have a neam that is greater than 164 pounds.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

$\mu = 171, \sigma = 34$

a. find the probability that if a person is randomly selected, his weight will be greater than 164 pounds.

This is 1 subtracted by the pvalue of Z when X = 164. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{164 - 171}{34}$

Z = -0.21

Z = -0.21 has a pvalue of 0.4168

1 - 0.4168 = 0.5832

58.32% probability that his weight will be greater than 164 pounds.

b. Find the probability that 12 randomly selected people will have a neam that is greater than 164 pounds.

Now $n = 12, s = \frac{34}{\sqrt{12}} = 9.81$

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{164 - 171}{9.81}$

Z = -0.71

Z = -0.71 has a pvalue of 0.2389

1 - 0.2389 = 0.7611

76.11% probability that 12 randomly selected people will have a neam that is greater than 164 pounds.

Answer from: Quest

answer: evaluate the objective function at each vertex to determine which - and -values, if any, maximize or minimize the function. example: find the minimum value and maximum value of the objective function f ( x , y ) = 4 x + 5 y , subject to the following constraints

Answer from: Quest

a and c

step-by-step explanation:

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