Question

Ayrshire cows produce an average of 57 pounds of milk per day with a standard deviation of 6 pounds. Assume the daily production is normally distributed. 20 cows are randomly selected from a herd and their day’s milk production is weighed. Let ȳ be the mean pounds of milk for this sample.

a) Find the mean and standard deviation for the sampling distribution of ȳ.

b) Use normalcdf with the sample mean ȳ to determine the probability that the mean weight of the milk produced by these cows is more than 58 pounds.

Answer 1

Solution :

Given that,

mean = = 57

standard deviation = = 6

n = 20

a) ȳ  = = 57

ȳ  = / n = 6 / 20 = 1.34

b) P(ȳ  > 58) = 1 - P(ȳ  < 58)

= 1 - P[(ȳ  - ȳ  ) / ȳ  < (58 - 57) / 1.34]

= 1 - P(z < 0.75)

Using z table,    

= 1 - 0.7734

= 0.2266