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8.5.5

(a) A voltage of constant, but unknown, value is to be measured. Each measurement X; is actually the sum of the desired voltage v and a noise voltage N, of zero mean and standard deviation of 1 (1-V), given by x = v +N, Assume that the noise voltages are independent random variables. i. Find the mean and variance of X;, i.e., E[X;) and VAR[X;). 2 5 ii. How many measurements are required so that the probability that the sample mean M. is within = 1 [1V) of the true mean is at least 0.997 (b) Suppose that the ith, i = 1, 2,..., 11, measurement of resistance is modeled by the random variable X;=r+N, with sample mean M., where r is the true resistance (unknown) and N is the noise random variable with zero mean and variance o i.e., NN (0,0%). i. Write the expression for (1-a) x 100% confidence interval in terms of number of measurements n, sample mean M., critical valueza/2, noise standard deviation on. Note that the parameter a (0,1). 3 ii. Let n = 100, ON = 10(9), M. = 330.2 (9). Find 98% confidence interval of the true resistance 5 (a) A voltage of constant, but unknown, value is to be measured. Each measurement X; is actually the sum of the desired voltage v and a noise voltage N, of zero mean and standard deviation of 1 (1-V), given by x = v +N, Assume that the noise voltages are independent random variables. i. Find the mean and variance of X;, i.e., E[X;) and VAR[X;). 2 5 ii. How many measurements are required so that the probability that the sample mean M. is within = 1 [1V) of the true mean is at least 0.997 (b) Suppose that the ith, i = 1, 2,..., 11, measurement of resistance is modeled by the random variable X;=r+N, with sample mean M., where r is the true resistance (unknown) and N is the noise random variable with zero mean and variance o i.e., NN (0,0%). i. Write the expression for (1-a) x 100% confidence interval in terms of number of measurements n, sample mean M., critical valueza/2, noise standard deviation on. Note that the parameter a (0,1). 3 ii. Let n = 100, ON = 10(9), M. = 330.2 (9). Find 98% confidence interval of the true resistance 5