Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

9.2. One-Way ANOVA 521

9.2.5.Using the notation of this section, assume that the means satisfy the con-
dition thatμ=μ 1 +(b−1)d=μ 2 −d=μ 3 −d=···=μb−d.Thatis,the
lastb−1 means are equal but differ from the first meanμ 1 , provided thatd =0.
Let independent random samples of sizeabe taken from thebnormal distributions
with common unknown varianceσ^2.

(a)Show that the maximum likelihood estimators ofμanddare ˆμ=X..and

dˆ=

∑b

j=2

X.j/(b−1)−X. 1

b

.

(b)Using Exercise 9.2.4, findQ 6 andQ 7 =cdˆ^2 so that, whend=0,Q 7 /σ^2 is

χ^2 (1) and

∑a

i=1

∑b

j=1

(Xij−X..)^2 =Q 3 +Q 6 +Q 7.

(c)Argue that the three terms in the right-hand member of part (b), once di-

vided byσ^2 , are independent random variables with chi-square distributions,

provided thatd=0.

(d)The ratioQ 7 /(Q 3 +Q 6 ) times what constant has anF-distribution, provided

thatd= 0? Note that thisFis really the square of the two-sampleTused to

test the equality of the mean of the first distribution and the common mean

of the other distributions, in which the lastb−1 samples are combined into

one.

9.2.6.On page 123 of their text, Kloke and McKean (2014) present the results of an

experiment investigating 4 drugs (treatments) for their effect on lowering LDL (low

density lipids) cholesterol. For the experimental design, 39 quail were randomly

assigned to one of the 4 drugs. The drug was mixed in their food, but, other than

this, the quail were all treated in the same way. After a specified period of time,

the LDL level of each quail was determined. The first drug was a placebo, so the

interest is to see if any other of the drugs resulted in lower LDL than the placebo.

Thedataareinthefilequailldl.rda. The first column of this matrix contains the

drug indicator (1 through 4) for the quail while the second column contains the ldl

level of that quail.

(a)Obtain comparison boxplots of LDL levels. Which drugs seem to result in

lower LDL levels? Identify, by observation number, the outliers in the data.

(b)Compute theF-test that all mean levels of LDL are the same for all 4 drugs.

Report theF-test statistic andp-value. Conclude in terms of the problem

using the nominal significance level of 0.05. Use the R code in Example 9.2.1.

(c)Does your conclusion in Part (b) agree with the boxplots of Part (a)?