91Ó°ÊÓ

Measurement errors from instruments are often modeled using the uniform distribution (see Problem 16). 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., \(-0.05 \leq x<0.03)\) ? (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.

Step by step solution

01

Define the Uniform Distribution

The measurement errors follow a uniform distribution over the interval \([-0.05, 0.05]\) microseconds. The probability density function (PDF) for a uniform distribution is flat across the interval, with a height of \(\frac{1}{b-a}\), where \(a = -0.05\) and \(b = 0.05\). This implies that the height of the PDF is \( \frac{1}{0.05 - (-0.05)} = 10 \).

02

Calculate the Probability for Part (a)

We need to find the probability that the measurement error \(x\) is less than \(+0.03\) microseconds, i.e., \(-0.05 \leq x < 0.03\). This is the interval's length \(0.03 - (-0.05) = 0.08\). Thus, the probability is \(0.08 \times 10 = 0.8\).

03

Calculate the Probability for Part (b)

The probability that the measurement errors are more than \(-0.02\) microseconds is the area from \(-0.02\) to \(+0.05\). The interval \(0.05 - (-0.02) = 0.07\) has a length of 0.07, so the probability is \(0.07 \times 10 = 0.7\).

04

Calculate the Probability for Part (c)

For the measurement error to be between \(-0.04\) and \(+0.01\) microsecond, the interval length is \(0.01 - (-0.04) = 0.05\). Thus, the probability is \(0.05 \times 10 = 0.5\).

05

Calculate the Mean and Variance for Part (d)

The mean of a uniform distribution over \([a, b]\) is \(\frac{a + b}{2}\). Here, \(a = -0.05\) and \(b = 0.05\), so the mean is \(\frac{-0.05 + 0.05}{2} = 0\). The variance is \(\frac{(b-a)^2}{12}\), where \((b-a) = 0.1\). Thus the variance is \(\frac{0.1^2}{12} = \frac{0.01}{12} \approx 0.0008333\), making the standard deviation \(\sqrt{0.0008333}\approx 0.02887\).

06

Conclusion on Unbiased Measurements

Since the mean of the measurement errors is zero, these acoustical sensors' measurements are considered unbiased, as unbiased measurements imply a mean error of zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Measurement Errors

Measurement errors are common in scientific and engineering measurements. They represent the discrepancies between the true value and the measured value. In the scenario described, we deal with time measurement errors, which occur due to the limitations of the measuring instrument. This error is modeled using a uniform distribution.

The uniform distribution describes a scenario where all outcomes in a given range are equally likely. In this exercise, our range is from \(-0.05\) to \(+0.05\) microseconds, which means the errors can spread equally throughout this range. Such a distribution helps us to calculate the probabilities of specific errors occurring.

• **Negative errors** imply that the measurement is perceived earlier than it truly is.
• **Positive errors** mean that the measurement is noted later than when it should have been recorded.

By understanding measurement errors, engineers can improve accuracy and account for potential biases in their results.

Probability Calculation

Calculating probabilities with a uniform distribution is simplified by its constant density function across an interval. Since all outcomes are equally likely, calculations often involve finding the portion of the interval that satisfies a certain condition.

The probability density function (PDF) for a uniform distribution between \(a\) and \(b\) has a height given by \(\frac{1}{b-a}\). In this instance, we have a height of \(10\) because the interval length is \(0.1\).
To find probabilities:

• **(a)** Find the portion of the interval less than \(+0.03\): The probability is calculated by multiplying the interval \([-0.05, 0.03)\) by the height (0.8).
• **(b)** For more than \(-0.02\): Evaluate the interval \([-0.02, 0.05]\), leading to a probability of \(0.7\).
• **(c)** Between \(-0.04\) and \(+0.01\), the probability is \(0.5\).

These calculations demonstrate the simplicity and efficiency of using uniform distribution assumptions for straight-forward probability evaluations.

Unbiased Measurements

An important factor in measurements is determining whether they are unbiased. Measurements are considered unbiased if the mean of the errors is zero. This implies that the errors are symmetrically distributed around zero, causing no systematic deviation from the true value.

In this exercise, the mean of the uniform distribution is calculated as \(\frac{a + b}{2}\), which in our case results in zero. This confirms the absence of bias in the measurements, as any deviation is equally likely to be too early or too late, fluctuating around the true measurement.

Understanding unbiased measurements helps in ensuring the reliability and accuracy of instruments. Engineers can trust that measurements are not inclined to default to error trends, thereby giving confidence in the outcomes derived from such data.