Question

Suppose that you theorize that the average ages of White Oaks, Quercus alba, in the following three locations. Site 1 Lower peninsula, Ml, ear Lake Michigan Site 2 Upper peninsula, MI, near Lake Superior and Site 3 Lower peninsula, Ml, 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? \(\mathrm{H}_{0}: \mu_{1}=\mu_{2}=\mu_{3} ; \mathrm{H}_{\mathrm{a}}\) : at least two of the means differ

\(\mathrm{H}_{0}: \mu_{1}=\mu_{2}=\mu_{3} ; \mathrm{H}_{\mathrm{a}}: \mu_{1} \neq \mu_{2} \neq \mu_{3}\)

(b) Which of the following conditions satisfied must be satisfied for using an ANOVA to analyze Dependence

Independence

Equal standard deviations

Nonequal standard deviations

\(\overline{\mathrm{X}}\) is normally distributed for each group

\(\overline{\mathrm{X}}\) is 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 \(\alpha=0.05\), what is your conclusion? There is insufficient evidence at the \(\alpha=0.05\) significance level to conclude that the mean age of White Oaks d

The data provides sufficient evidence at the \(\alpha=0.05\) significance level to conclude that the mean age of White

Answer 1

a.) H₀: µ₁ = µ₂ = µ₃; H_a: at least two of the means differ

   Because the question says "are all not the same"

b.) Independence

Equal Standard Deviations

X-bar Normally distributed for each group

c.) p-value = 0.2241

d.) p-value > 0.05. Hence, we can not reject the null hypothesis.

So, there is insufficient evidence to at α = 0.05 significance level to conclude that the mean age of the white oak tress differ between at least two sites