数理统计 英文 Homework_12

Homework 11
(Due on June 11th )
1. Specimens of milk from a number of dairies in three different districts were analyzed, and the concentration of the radioactive isotope strontium-90 was measured in each specimen. Suppose that specimens were obtained from four dairies in the first distinct, from six dairies in the second district, and from three dairies in the third district; and the results measured in picocuries per liter were as follows:
District 1: 6.4, 5.8, 6.5, 7.7,
District 2: 7.1, 9.9, 11.2, 10.5, 6.5, 8.8 District 3: 9.5, 9.0, 12.1
a. Assuming that the variance of the concentration of strontium-90 is the same for the dairies in all three dairies in all three districts, determine the M.L.E. of the mean concentration in each of the districts and the M.L.E. of the common variance.
b. Test the hypothesis that the three districts have identical concentrations of strontium-90. 2. Show that in a one-way layout the following statistic is an unbiased estimator of 2
σ:
211
1()i
n p i ij i j Y Y n p +==--∑∑
3. In 1973, the president of Texaco, Inc., made a statement of to a U.S. Senate subcommittee concerned with air and water pollution. The committee was concerned with, among other things, the noise levels associated with automobile filters. He cited the data in Table 1 from a study that included vehicles of three different sizes.
Table 1
Vehicle size
Noise values Small 810, 820, 820, 835, 835, 835 Medium 840, 840, 840, 845, 855, 850 Large
785, 790, 785, 760, 760, 770
a. Construct the ANOV A table for this data.
b. Compute the p -value fir the null hypothesis that all three sizes of vehicles produce the same
level of noise on average.
4. Consider a two-way layout in which the values of ()ij E Y for 1,...,i I = and 1,...,j J = are as given in each of the following four matrices. For each matrix, state whether the effects of the factors are additive.
a. b. Factor B Factor A
1 2 1 5 7 2
10
14
c. d.
Factor B Factor A 1 2 3 4 1 3 -1 0 3 2 8 4 5 8 3 4
1
4
5. Suppose that in a two-way layout, with 2I = and 2J =, the values of ()ij E Y are as given in part (b) of Exercise 4. Determine the values of μ, 1α, 2α, 1β and 2β that satisfy
1
0I
i
i α
==∑,
1
0J
j
j β
==∑ and ()ij i j E Y μαβ=++ for 1,...,i I = and 1,...,j J =.
6. In a study to compare the reflective properties of various paints and various plastic surfaces, three different types of paint were applied to specimens of five different types of plastic surfaces. Suppose that the observed results in appropriate coded units were as shown in Table 2.
Table 2
Type of surface
Type of paint 1 2 3 4 5
1 14.5 13.6 16.3 23.
2 19.4 2 14.6 16.2 14.8 16.8 17.
3 3 16.2 14.0 15.5 18.7 21.0
Test the hypothesis that the reflective properties of the three different types of paint are the same.
Now we have completed all the homework within lecture plan. Good luck to everybody in your final exam!
Factor B Factor A
1 2 1 3 6 2
4
7
Factor B Factor A
1 2 3 4 1 1 2 3 4 2 2 4 6 8 3
3
6
9
12。