please help me solve this

Question

On the distant planet Cowabunga , the weights of cows have a normal distribution with a mean of 569 pounds and a standard deviation of 59 pounds. The cow transport truck holds 17 cows and can hold a maximum weight of 10370. If 17 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 10370? (This is the same as asking what is the probability that their mean weight is over 610.)

Answer 1

Since the population standard deviation is 59 pounds, the variance is 59*59 = 3,481.

For sums of random variables, both means and variances are summed. For a sample size of 17, the sample mean is 17 * 569 = 9,673 and the sample variance is 17 * 3,481 = 59,177. The sample standard deviation is the square root of the sample variance, or 243.26.

The Z value is (10370 - 9,673)/243.26 = 697/243.26 = 2.8652.

This is a one-tailed test, as what is being sought is the likelihood that the weight of the sample of 17 exceeds 10,370.

Using a standard normal cumulative distribution table, the values for both 2.86 and 2.87 are .9979 to four decimal places.

The probability that the sample exceeds 10,370 pounds in weight is 1-0.9979 = 0.0021, or about one chance in 480.