On the distant planet Cowabunga , the weights of cows have a normal distribution with a mean of 459 pounds and a standard deviation of 61 pounds. The cow transport truck holds 16 cows and can hold a maximum weight of 7616. If 16 cows are randomly selected from the very large herd to go on the truck, what is the probability their total weight will be over the maximum allowed of 7616? (This is the same as asking what is the probability that their mean weight is over 476.)

13.35% probability their total weight will be over the maximum allowed of 7616

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:

If 16 cows are randomly selected from the very large herd to go on the truck, what is the probability their total weight will be over the maximum allowed of 7616?

7616/16 = 476

So this is 1 subtracted by the pvalue of Z when X = 476.

By the Central Limit Theorem

has a pvalue of 0.8665

1 - 0.8665 = 0.1335

13.35% probability their total weight will be over the maximum allowed of 7616

answer: can you me with homework?

step-by-step explanation: