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November 30th, 2006, 06:02 AM   #1
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Statistics: Two designs to obtain estimates

The weights of three individual objects are to be estimated. Two different designs to obtain these estimates are being considered:

D1:
The three objects are weighed individually, seven times each.

D2:
i. Each object is weighed separately once.
ii. Then the objects are weighed two at a time in the three possible, different combinations.
iii. Finally, the three objects are weighed all together, once.
iv. This factorial design (steps i, ii, and iii) is repeated three times.

The two designs both use all of the n = 21 measurements for which funds are available.

Which design do you EXPECT to yield the more precise estimates of the individual weights of the three objects? Why? Show the details of your work.

What assumptions does your answer require?

Note: To answer this question you do no need to know, or use, the "y" values; nor do you need to estimate the residual mean squares.

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Please help me solve this problem. If you know any websites that would be helpful, please tell me.
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December 11th, 2006, 03:08 PM   #2
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D2 wins

I would think that D2 would be better. Assume that you can measure to tenths of a gram. Sig digs allow you to measure halves of decigrams. By this, you have the individual measurements. When you weigh multiple weights at the same time however, you can find halves and thirds of whaterever your estimates were. Say you have weights A, B, and C and the measurements:

A=3.35 g, 3.4 g, and 3.45 g
B=5.6 g, 5.6 g, 5.6 g
C=1.0 g, 0.95 g, 1.0 g
A+B=9.0g, 8.95 g, 8.95 g
A+C=4.3 g, 4.3 g, 4.35 g
B+C=6.5 g, 6.55 g, 6.55 g
A+B+C=9.85 g, 9.85 g, 9.85 g

You can calculate that:
A=3.34 g
B=5.60 g
C=0.93 g

With guaranteed accuracy of two of three hundredths of a gram.

While D1 only offers certainty to five hundredths of a gram.
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