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.

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 and standard deviation , the zscore of a measure X is given by:

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 and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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:

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

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

By the Central Limit Theorem

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.

my answer was deleted because it was to vague the answer is 12.76 becuase its in the hundreths place. john

1. 37

2. 22

3. 44

4. 66

step-by-step explanation: