Suppose that you theorize that the average ages of White Oaks, Quercus alba, in the following three locations.

Site 1 : Lower peninsula, MI, near Lake Michigan

Site 2 : Upper peninsula, MI, near Lake Superior and

Site 3 : Lower peninsula, MI, near Saginaw Bay (are not all the same). To test this theory, you take 3 simple random samples of 35 trees from each site, and record the ages of each White Oak in each sample. (a) What are the hypotheses?

H₀: ?₁= ?₂= ?₃vs. H_a: At least two of the means differ from each other.

H₀: ?₁= ?₂= ?₃vs. H_a: At least two of the means are different from the third.

H₀: ?₁= ?₂= ?₃vs. H_a: At least one of the means is different from all the others.

H₀: ?₁= ?₂= ?₃; H_a: ?₁? ?₂? ?₃

(b) Which of the following conditions satisfied must be satisfied for using an ANOVA to analyze this data set? Select all that apply.

Dependence

Independence

Equal standard deviations

Nonequal standard deviations

Xis normally distributed for each group

Xis normally distributed for at least one of the groups (c) What is the p-value of the ANOVA test? Use 4 decimal places. (d) If ? = 0.05, what is your conclusion?

There is insufficient evidence at the ? = 0.05 significance level to conclude that the mean age of White Oaks differs at between at least two sites.

The data provides sufficient evidence at the ? = 0.05 significance level to conclude that the mean age of White Oaks is the same for all three sites.

The data provides sufficient evidence at the ? = 0.05 significance level to conclude that the mean age of White Oaks differs between at least two sites.

(e) What does the p-value tell us?

The probability we would find sample means at least as far apart as we did, assuming the population means are all equal.

The probability the population means for all three groups are equal, assuming the sample means are as given above. The probability the population means are different for at least one group, assuming the sample means are as given above.

The probability we would find sample means at least as close together as they are, assuming the population means are all equal.

(f) Suppose the mean age of White Oaks in the Lower peninsula were 200, the mean age of all White Oaks in the Upper peninsula were 300, and the mean age of all White Oaks in the Lower peninsula were 200. Based on your conclusion in the previous problem, you have made

a Type I error.

a Type II error.

no error at all.

Histogram of Age (Site 1) Histogram of Age (Site 2) Histogram of Age (Site 3) 14 12 10 Frequency 8 Frequency Frequency Box Plot of Comparison of Age F. 40.00 - 60.00 Site 3 80.00 20.00 120.DD 180.00 100.00 160.00 140.00 D.DO + 200.00 220.00 40.00 60.00 80.00 100.00 50.00 140.00 120.00 160.00 180.00 200.00 220.00 150.00 100.00 200.00 250.00 Q-Q Normal Plot of Age(Site 1) 0-Q Normal Plot of Age(Site 2) 0-Q Normal Plot of Age(Site 3) Site 2 3.5 3.5 3.5 Site 1 2.5 2.5 2.5 - 1.5 1.5 1.5 - 50 100 150 200 250 0.5 0.5 0.5 Standardized Q-Value Standardzed Q-Value Standardzed Q-Value 3.5 -1.5 -DAY 10.5 25 3.5 -15 LOVE 0.5 2.5 3.5 15 0 0.5 2.5 -1.5 1.5 -2.5 -2 5 - -2.5 -3.5 -3 5 -3.5 Z- Value Z-Value Z-ValueANOVA Summary Total Sample Size 105 Grand Mean 107.84 Pooled Std Dev 45.44 Pooled Variance 2064.60 Number of Samples 3 Confidence Level 95.00% ANOVA Sample Stats Site 1 Site 2 Site 3 Sample Size 35 35 35 Sample Mean 118.74 101.80 102.97 Sample Std Dev 37.36 54.33 42.96 Sample Variance 1396.02 2951.93 1845.85 Pooling Weight 0.3333 0.3333 0.3333 Sum of Degrees of Mean F-Ratio OneWay ANOVA Table Squares Freedom Squares p-Value Between Variation 6266.99 2 3133.50 1.52 0.2241 Within Variation 210589.26 102 2064.60 Total Variation 216856.25 104