STA 106 Homework 2 Solutions

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STA 106 Homework 2 Solutions


Book Problems

1.  (a)  The estimated values of µi  are simply the group means, so the estimated value for µ 1  is 38, for µ2  is 32, and for µ3  is 24.

(b) We know E{Yij } = µi  for all i, j, so that the estimated expected values are the group means.  In other words, we estimate the average value of Y to be 38 for all subjects in group I, 32 for all subjects in group II, and 24 for all

subjects in group III.

(c)  SSTO = sY(2)(nT  − 1) = (6.88)2 (24 − 1) = 1088.6912

SSA =     i ni (i ] − ]] )2  = (8)(38 − 32)2 + (10)(32 − 32)2 + (6)(24 − 32)2  = 288 + 0 + 384 = 672

SSE = SSTO − SSA = 1088.6912 − 672 = 416.6912

(d)  d.f.{SSTO} = nT  − 1 = 24 − 1 = 23 d.f.{SSA} = a − 1 = 3 − 1 = 2

d.f.{SSE} = nT  − a = 24 − 3 = 21

(e)  MSTO = = = 47.3344

MSA = = = 336

MSE = = = 19.8424

2. (a)

Since one standard deviation is almost twice that of another, it seems unreasonable to assume the variance is constant

for each group.

(b)

The fastest recovery time is group III (the “above average” exercise group). This group is more than two standard deviations away from the other groups (at least), so it may be reasonable to conclude it is significantly different.

(c)

(d)

H0 : µI = µI I = µI I I vs. HA At least one population average is different (not equal).

FS = = = 16.9334.

The p-value is :Pr{F > 16.9334} = 0.0000417. By the F table, it is approximately < 0.0001.

(e)

We reject the null hypothesis, and conclude that at least one true average recovery time (in days) is significantly different between the groups I, II, III.

Or, we cannot conclude that the true average recovery time is the same for all groups.

3. (a)

We estimate the values using the sample means, so that the estimates for µi − µ ] are (for groups M, O, Y respectively):

27.75 - 23.56 = 4.19, 21.42 - 23.56 = -2.14, and 21.5 - 23.56 = -2.06.

(b)

These represent the difference in sample means from the overall mean. For example, the Y group tends to be -2.06 lower than the overall average.










2022-05-25