MTH205 Introduction to Statistical Methods
Tutorial 5
Based on Chapter 5
1. Show that in the one-way analysis of variance, the error mean square MSE is an unbiased estimator of the treatment variance σ2. Hint: You may wish to start from the definition of SSE, i.e.
we first add and subtract the treatment mean µi from each term to obtain
and then expand.
2. For the one-way ANOVA table, we know that
Show that SST = SSTr + SSE.
3. Show that in the one-way analysis of variance,
E (SSTr)=(I − 1) σ2 if H0 is true
E (SSTr) > (I − 1) σ2 if H0 is false.
You should start from the definition of
4. The removal of ammoniacal nitrogen is an important aspect of treatment of leachate at landfill sites. The rate of removal (in percent per day) is recorded for several days for each of several treatment methods. The results are presented in the following table.
Construct the one-way ANOVA table. Do the treatment methods di↵er in their rates of removal?
5. An experiment was performed to determine whether the annealing temperature of ductile iron affects its tensile strength. Five specimens were annealed at each of four temperatures. The tensile strength (in ksi) was measured for each. The results are presented in the following table.
(i) Construct the one-way ANOVA table. Can you conclude that there are di↵erences among the mean strengths?
(ii) Use the Bonferroni method to determine which pairs of means, if any, are di↵erent at the 5% significance level.
(iii) Use the Tukey-Kramer method to determine which pairs of means, if any, are di↵erent at the 5% significance level.
(iv) Which is the more powerful method to find all the pairs of treatments whose means are different, the Bonferroni method or the Tukey-Kramer method?
A metallurgist wants to determine whether the mean tensile strength for specimens annealed at 900◦C differs from the mean strengths for specimens annealed at 750◦C, 800◦C and 850◦C.
(v) Use the Bonferroni method to determine which pairs of means, if any, for 750◦C, 800◦C and 850◦C to differ from the mean for 900◦C at the 5% significance level.
(vi) Use the Tukey-Kramer method to determine which pairs of means, if any, for 750◦C, 800◦C and 850◦C to differ from the mean for 900◦C at the 5% significance level.
(vii) Which is the more powerful method to find all the pairs of treatments whose means di↵er from the 900◦C, the Bonferroni method or the Tukey-Kramer method?