Question

The following are some of the data collected in one of the mensuration exercises. The measurements are volumes in m3/ha for 24 different plots:

Column 1	Column 2
736	1009
1560	1413
990	1528
924	1079
1042	1207
1360	1770
936	826
1618	1573
1221	1259
1033	1564
1288	896
555	1462

a) Assume that each column of data comes from a different forest type. Is it very likely to get a variance ratio S₁²/S₂² as big as, or greater than what you got if you assume that ?₁² = ?₂² . Can you assume that ?₁²= ?₂² with a good probability?

b)You don't know the population variance of the two forest types, but you are told that µ₁ = µ₂ . What is the probability that, if you take 12 plots from each type:

i)x?₂ - x?₁ ? 200?

ii) 100 ? x?₂ - x?₁ ? 200?

iii) to get a difference as big as you have or greater?

Answer 1

A)

	Column 1	Column 2
	736	1009
	1560	1413
	990	1528
	924	1079
	1042	1207
	1360	1770
	936	826
	1618	1573
	1221	1259
	1033	1564
	1288	896
	555	1462
Mean	1105.25	1298.833333
standard deviation	316.8774139	299.6740148
Var	100411.2955	89804.51515

Null and Alternate Hypothesis

Level of significance alpha = 0.05

A way to compare the variances of two normally distributed populations is to use the variance ratio. The sampling distribution of is used. Since the population variances are usually not known, the sample variances are used. The assumptions are that sample variances s_1^2 and s_2^2 are computed from independent samples of size n1 and n2, respectively, drawn from two normally distributed populations. If the assumptions are met, follows a distribution known as the F distribution with two values used for degrees of freedom.

95% Confidence interval of the ratio of population variances can be calculated as follows:

Thus we conclude that the ratio of population variance lies between 0.32 and 3.85 with 95% confidence interval.

B) i)

Lookup up the standard normal distribution table

P(X<=3.126) = 0.941

ii)

P(z <= 2.331) = 0.940

P(z <= 3.126) = 0.941