In a sophisticated medicine research center, two scales are used for repeated weighing of the same item. If the true weight of a compound is 2 grams [g], the first scale produces readings X that have a mean 2.0 g and standard deviation 0.03 g. The second scale’s readings Y have mean 2.001 g and standard deviation 0.01 g.
a) Which scale is more appropriate for the measurement?
b) What are the mean and standard deviation of the difference Y-X between the readings? (The readings X and Y are independent)
c) You measure once with each scale and average your readings. Your result is Z=½(X+Y). What are μZ and σZ? Is the average Z more or less variable than the reading Y of the less variable scale?
a) To determine which scale is more appropriate for the measurement, we can compare the standard deviations of the readings from each scale. A smaller standard deviation indicates less variability in the measurements, which is generally desirable.
In this case, the first scale has a standard deviation of 0.03 g, while the second scale has a standard deviation of 0.01 g. Therefore, the second scale is more appropriate for the measurement because it has a smaller standard deviation, indicating less variability in the readings.
b) To find the mean and standard deviation of the difference Y-X between the readings, we can use the properties of independent random variables.
The mean of the difference is given by the difference of the means: μY-X = μY - μX = 2.001 g - 2.0 g = 0.001 g.
The variance of the difference is calculated as the sum of the variances: Var(Y-X) = Var(Y) + Var(X) = (0.01 g)^2 + (0.03 g)^2.
The standard deviation is the square root of the variance: σY-X = √[Var(Y) + Var(X)] = √[(0.01 g)^2 + (0.03 g)^2].
c) To find the mean and standard deviation of the average Z = 1/2(X+Y), we need to consider the properties of the sum and scalar multiplication of random variables.
The mean of Z is given by μZ = 1/2(μX + μY) = 1/2(2.0 g + 2.001 g) = 2.0005 g.
The variance of Z is calculated as follows:
Var(Z) = 1/4[Var(X) + Var(Y) + 2Cov(X,Y)], where Cov(X,Y) represents the covariance between X and Y.
Assuming independence, Cov(X,Y) = 0. Therefore, Var(Z) = 1/4[Var(X) + Var(Y)] = 1/4[(0.03 g)^2 + (0.01 g)^2].
The standard deviation is the square root of the variance: σZ = √[Var(Z)] = √[1/4[(0.03 g)^2 + (0.01 g)^2]].
To compare the variability of Z with the less variable scale's reading Y, we can compare their standard deviations. If σZ is smaller than σY, it indicates that Z is less variable. If σZ is larger than σY, it indicates that Z is more variable.
If we calculate the standard deviation of Z and compare it with the standard deviation of Y, we can determine whether Z is more or less variable.