Uniform Distribution: Measurement Errors Measurement errors from instruments are often modeled using the uniform distribution (see Problem 12). To determine the range of a large public address system, acoustical engineers use a method of triangulation to measure the shock waves sent out by the speakers.

The time at which the waves arrive at the sensors must be measured accurately.

In this context, a negative error means the signal arrived too early. A positive error means the signal arrived too late. Measurement errors in reading these times have a uniform distribution from 0.05 to 0.05 microseconds.

(Reference: J. Perruzzi and E. Hilliard, “Modeling Time Delay Measurement Errors,” Journal of the Acoustical Society of America, Vol. 75, No. 1, pp.

197–201.) What is the probability that such measurements will be in error by

(a) less than 0.03 microsecond (i.e., 0.05  x 0.03)?

(b) more than 0.02 microsecond?

(c) between 0.04 and 0.01 microsecond?

(d) Find the mean and standard deviation of measurement errors.

Measurements from an instrument are called unbiased if the mean of the measurement errors is zero. Would you say the measurements for these acoustical sensors are unbiased? Explain.