Question

An engineer measures a quantity q. It is assumed there is a random error in each measurement, so the engineer takes n measurements and reports the average of the measurements as the estimated value of q. If $Y_i$ is the $i^{th}$ measure of q, assume that $Y_i = q + X_i$, where $X_i$ is the error in the $i^{th}$ measurement. Also assume that the $X_i$'s are independent and identically distributed (i.i.d.) random variables with mean $E(X_i) = 0$, and variance $V(X_i) = 4$. The engineer reports the average of measurements $M_n = frac{Y_1 + Y_2 + ... + Y_n}{n}$. How many measurements does the engineer need to make, to be 95% certain that the error in $M_n$ is less than 0.1? In other words, what is the minimum value of n for which $P(q - 0.1 le M_n le q + 0.1) ge 0.95$? Explain how your answer is consistent with the assumptions of the calculation.

          An engineer measures a quantity q. It is assumed there is a random error in each measurement, so the engineer takes n measurements and reports the average of the measurements as the estimated value of q.
If $Y_i$ is the $i^{th}$ measure of q, assume that $Y_i = q + X_i$, where $X_i$ is the error in the $i^{th}$ measurement. Also assume that the $X_i$'s are independent and identically distributed (i.i.d.) random variables with mean
$E(X_i) = 0$, and variance $V(X_i) = 4$. The engineer reports the average of measurements $M_n = frac{Y_1 + Y_2 + ... + Y_n}{n}$.
How many measurements does the engineer need to make, to be 95% certain that the error in $M_n$ is less than 0.1? In other words, what is the minimum value of n for which $P(q - 0.1 le M_n le q + 0.1) ge 0.95$?
Explain how your answer is consistent with the assumptions of the calculation.

$An engineer measures a quantity q. It is assumed there is a random error in each measurement, so the engineer takes n measurements and reports the average of the measurements as the estimated value of q. If Yi is the i^th measure of q, assume that Yi = q + Xi, where Xi is the error in the i^th measurement. Also assume that the Xi's are independent and identically distributed (i.i.d.) random variables with mean E(Xi) = 0, and variance V(Xi) = 4. The engineer reports the average of measurements Mn = fracY1 + Y2 + ... + Ynn. How many measurements does the engineer need to make, to be 95% certain that the error in $Mn$ is less than 0.1? In other words, what is the minimum value of n for which $P(q - 0.1 le Mn le q + 0.1) ge 0.95$? Explain how your answer is consistent with the assumptions of the calculation.$

Added by James W.

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