Question

A sheep farmer suspects that when their animals are given a feed with a higher nitrogen content they tend to produce a greater weight of wool. To test this claim the farmer gave a random selection of 18 of their sheep a low nitrogen content feed and gave another random selection of 13 of their sheep a high nitrogen content feed. In the low nitrogen group it was found that the sheep produced an average of 12.5 pounds of wool with a standard deviation of 2.4 pounds. In the high nitrogen group it was found that the sheep produced an average of 16.2 pounds of wool with a standard deviation 3.1 pounds.

(a) Use these results to calculate a 95% confidence interval for the true mean difference in wool weight produced between the two groups. (You may order your two groups however you wish, but make your choice clear.) If needed, you may assume that the weight of wool produced under both types of feed is normally distributed.

(b) Using your confidence interval from part (a), is it reasonable to say that the weight of wool produced between the low nitrogen group and the high nitrogen group is the same? Explain.

Answer 1

Let group 1 be low nitrogen content group and group 2 be high nitrogen content group.

Two random samples are given with different subjects. These are therefore independent samples. We want to test for the difference of the two populations as a whole. We assume that the data is normal and the unknown population variances are equal. We will conduct an independent samples t-test for difference of population means with equal variances.

	Group 1	Group 2
n	18	13
Mean	12.5	16.2
SD	2.4	3.1
Var	5.76	9.61

(a) Use these results to calculate a 95% confidence interval for the true mean difference in wool weight produced between the two groups. (You may order your two groups however you wish, but make your choice clear.) If needed, you may assume that the weight of wool produced under both types of feed is normally distributed.

(1- $1596222811546_blob.png$ )% CI for population difference

(11 – 12) Ftmı +92-2,0/2 s (1/n1+1/n2))

Where sample mean differnce = -3.7

Pooled variance S_p² =

Vari(nı – 1) + Varz(n2 - 1) ni + n2 - 2

=7.3531

$1596222811546_blob.png$ = 1 - 0.95 = 0.05

C.V. =

tnl+n2-2,0/2

=t_{29, 0.025}

C.V. = 2.0452

MOE =
tnl+n2-2,0/2
*
sz(1/n1+1/n2)

Subsituting the values

Ans:
(-5.7186 < μl – μ2 < -1.6814)

(b) Using your confidence interval from part (a), is it reasonable to say that the weight of wool produced between the low nitrogen group and the high nitrogen group is the same? Explain.

It is not reasonable to say that the weight of wool produced between the low nitrogen group and the high nitrogen group is the same.

Reason: The confidence interval is interpreted as we are 95% confident that the true mean difference lies within this interval. Since the interval does not contain '0' means there is very low possibility (0.05) of having no difference at all.