Question

Do eggs from 27 week old hens weigh the same on average as eggs from 54 week old hens? We will study this question by taking a random sample of m=31 eggs from 27 week old hens and n=24 eggs from 54 week old hens. The mean weights were x= 54.6 grams for the 27 week old hens and y=72.7 grams inches for the eggs from 54 week old hens. It is recognized that the true standard deviation of 27 week old hen eggs is ?x = 4.4 grams and ?y = 4.8 grams for the 54 week old hen eggs. The true (unknown) mean weight of 27 week old hen eggs = ?x , while the true unknown mean weight of 27 week old hen eggs is ?y for the males.
a) What is the standard deviation of the distribution of X?
b) What is the standard deviation of the distribution of X - Y ?
c) Create a 98% confidence interval for ?x ? ?y ? ,
d) What is the length of the confidence interval in part c) ?

Answer 1

a)

Std.deviation of X = s/sqrt(n)
= 4.4/sqrt(31)
= 0.7903

b)

standard deviation of the distribution of X - Y = sqrt(s1^2/n1 + s2^2/n2) = sqrt(4.4^2/31 + 4.8^2/24)
= 1.2588

c)
z value at 98% = 2.3263

CI = ( x1 -x2) + /- z *sqrt(s1^2/n1 + s2^2/n2)
= (54.6 - 72.7) +/- 2.3263*sqrt(4.4^2/31 + 4.8^2/24)
= (-21.0283,-15.1717)

98% confidence interval for ?x ? ?y = (-21.0283,-15.1717)

d)

the length of the confidence interval = -15.1717 + 21.0283
= 36.2