Question

.  The Pomme d’Alouette Orchard provides apples to a national maker of “hard” apple cider.  The recent growth of consumer demand for cider has encouraged the orchard to plant new apple trees in order to increase substantially its production of apples.

The orchard is considering the purchase of a new variety of apple tree.  NeuApfel, developers of the new variety, claim a square acre planted in the new variety will produce the same average yield (in tons) of apples each year as the orchard’s current variety and will do so at a lower cost.

To evaluate the claim that the new variety of apple trees will produce the same average yield of apples, the orchard and NeuApfel compare the yields (in tons) for the past 16 harvests:

Orchard variety: 209, 200, 188, 196, 202, 179, 198, 203, 185, 207, 205, 194, 189, 201, 195, 192

NeuApfel variety: 199, 195, 199, 184, 206, 180, 193, 188, 197, 205, 181, 190, 186, 195, 188, 183

In what follows, assume that the yield of apples produced for each variety of trees is normally distributed, and that the variances are unknown but equal.

(a)  Use the appropriate two-population test to determine at the 5% level of significance whether the average yields for the two varieties are equal or not.

(b)  Use the Analysis of Variance technique to determine at the 5% level of significance whether the average yields for the two varieties are equal or not.

Answer 1

Two-Sample T-Test and CI: Orchard variety, NeuApfel variety

ans a ) claim:the new variety of apple trees will produce the same average yield of apples

let us consider

using minitab>stat>basic stat> two sample t test we have

Two-sample T for Orchard variety vs NeuApfel variety

N Mean StDev SE Mean
Orchard variety 16 196.44 8.33 2.1
NeuApfel variety 16 191.81 8.12 2.0

Difference = μ (Orchard variety) - μ (NeuApfel variety)
Estimate for difference: 4.63
95% CI for difference: (-1.32, 10.57)
T-Test of difference = 0 (vs ≠): T-Value = 1.59 P-Value = 0.122 DF = 30
Both use Pooled StDev = 8.2267
since p value 0.122 is greater than 0.05 so we accept H0

we have sufficient evidence to accept the claim that the new variety of apple trees will produce the same average yield of apples.

Ans b )

One-way ANOVA: Orchard variety, NeuApfel variety

Method

Null hypothesis All means are equal
Alternative hypothesis At least one mean is different
Significance level α = 0.05

Equal variances were assumed for the analysis.

Factor Information

Factor Levels Values
Factor 2 Orchard variety, NeuApfel variety

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Factor 1 171.1 171.13 2.53 0.122
Error 30 2030.4 67.68
Total 31 2201.5

Model Summary

S R-sq R-sq(adj) R-sq(pred)
8.22673 7.77% 4.70% 0.00%

Means

Factor N Mean StDev 95% CI
Orchard variety 16 196.44 8.33 (192.24, 200.64)
NeuApfel variety 16 191.81 8.12 (187.61, 196.01)

Pooled StDev = 8.22673
since p value is greater than 0.05 so we can conclude that the average yields for the two varieties are equal