Question

On the distant planet Cowabunga , the weights of cows have a normal distribution with a mean of 506 pounds and a standard deviation of 84 pounds. The cow transport truck holds 10 cows and can hold a maximum weight of 5430. If 10 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 5430?

Answer 1

The probability that the total weight of 10 cows exceeds 5430 pounds is determined by calculating the z-score for 5430 pounds using the mean and standard deviation, then finding the corresponding probability on the standard normal distribution.

Step-by-step explanation:

To calculate the probability that the total weight of 10 cows on the planet Cowabunga exceeds the maximum allowed weight of 5430 pounds, we first need to understand the distribution of the total weight of 10 randomly selected cows. Since we know the weight of cows follows a normal distribution with a mean (μ) of 506 pounds and a standard deviation (σ) of 84 pounds, we can use the Central Limit Theorem to find the distribution of the sample sum.

The mean of the sample sum is the sum of the means, which is 506 pounds × 10 cows = 5060 pounds. The standard deviation of the sample sum is the standard deviation times the square root of the sample size, which is 84 pounds × √10. To find the probability that the total weight exceeds 5430 pounds, we calculate the z-score for 5430 pounds using the formula:

Z = (X - μ√N) / (σ√N), where X is 5430 pounds, μ is 506, √N is 10, and σ is 84.

Once we have the z-score, we can use the standard normal distribution table (or a calculator with the normal distribution function) to find the area to the right of that z-score, which represents the probability that the sum weight of 10 cows exceeds 5430 pounds.