Suppose that an object is 10 inches long and suppose that a measuring device giv

Question

Suppose that an object is 10 inches long and suppose that a measuring device gives the readings   7 inches with probability 1/4,    9 inches with probability 1/4,   11 inches with probability 1/4 and 13 inches with probability 1/4.  

In 16 rows Write down all 16 possible ways that two measurements can turn out along with the average of these two measurements. Use these 16 possibilities to compute the distribution of x when n = 2(i.e., find P (x=7), P (x =8) etc.). The average is most accurate when a high measurement is paired with a low measurement. Go back to the 16 possible rows and put a check mark by those possibilities where one measurement of the two measurements is above 10 and the other of the two measurements is below 10, that is where X1 < 10 and X2 > 10 or X1 > 10 and X2 < 10. i.e. where high measurements tend to cancel low measurements.

Conduct the experiment by measuring the object twice, then average the two measurements to estimate the length of the object. This means find your value for the first measurement X1, and find your value for the second measurement X2, then average these two measurements. Go back to the 16 possibilities and place an arrow pointing to which one of the 16 possible rows came up in your Experiment [You can the experiment by rolling a die and if a 1 comes up assume the measurement was 7, if a 2 comes up assume the measurement was a 9, etc.   If 5 or 6 comes up roll a die again.]

Explanation / Answer

From the given data,

From table,

P(Xbar = 7) = 1/16

P(Xbar = 8) = 2/16= 1/8

P(x1>10 and X2<10) = 4/16 = 1/4

P(X<10 and X2>10) = 4/16 = 1/4

X1 X2 Mean 7 7 7 7 9 8 7 11 9 7 13 10 9 7 8 9 9 9 9 11 10 9 13 11 11 7 9 11 9 10 11 11 11 11 13 12 13 7 10 13 9 11 13 11 12 13 13 13

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