Problem 64 - An Introduction to Mathematical Statistics and Its Applications

Question

A circuit contains three resistors wired in series. Each is rated at 6 ohms. Suppose, however, that the true resistance of each one is a normally distributed random variable with a mean of 6 ohms and a standard deviation of \(0.3\) ohm. What is the probability that the combined resistance will exceed 19 ohms? How "precise" would the manufacturing process have to be to make the probability less than \(0.005\) that the combined resistance of the circuit would exceed 19 ohms?

Step-by-Step Solution

Answer

The probability that the combined resistance of the circuit will exceed 19 ohms is approximately 0.0287. To make this probability less than 0.005, the standard deviation in the resistance of each resistor would have to be about 0.364 ohms.

Step 1: Calculate combined mean and standard deviation

The mean of the combined resistance is the sum of the individual means, which is 3 times 6 ohms, equal to 18 ohms. The standard deviation of a sum of independent random variables is the square root of the sum of their variances. Since standard deviation of one resistor is 0.3 ohms, and variance is (standard deviation)², the variance of one resistor is \(0.3^2 = 0.09\). The combined variance is 3 times this, i.e., 0.27, which makes the combined standard deviation \(\sqrt{0.27}=0.5196\) ohms.

Step 2: Calculate the Z score

The Z score is the number of standard deviations a particular value is from the mean. It is calculated as \(Z = (X - \mu) / \sigma\), where is the value in question (19 ohms), \(\mu\) is the mean (18 ohms), and \(\sigma\) is the standard deviation (0.5196 ohms). The Z score hence is \(Z = (19 - 18) / 0.5196 = 1.924\).

Step 3: Calculate the probability

To find the probability that the combined resistance is greater than 19 ohms, we look up this Z value in the Z-table, which gives the cumulative probability. The cumulative probability for a Z score of 1.924 is 0.9713. The desired probability is therefore 1 - 0.9713, or 0.0287.

Step 4: Determine precision level for the operation

In order to limit the probability to less than 0.005, we need to adjust the standard deviation since decreasing the variability will decrease the chance of extreme outcomes. From Z-tables, a Z value of approximately 2.576 corresponds to a cumulative probability of (1 - 0.005). Hence, the upper limit for the combined resistance under this new precision level would still be 19 ohms, the mean would still be 18 ohms, but we need to solve the following for the standard deviation, \(\sigma\): \(Z = (19 - 18) / \sigma = 2.576\). Solving for \(\sigma\) we get \(\sigma = (19 - 18) / 2.576 = 0.388\). Hence, the standard deviation of the resistance of each resistor would have to be \(\sqrt{0.388 / 3} = 0.364\) ohms.

Key Concepts

Normal Distribution

When learning about probability and statistics, one of the most important concepts to grasp is the normal distribution, sometimes referred to as the Gaussian distribution or bell curve. This distribution is a continuous probability distribution that is symmetric around its mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

In practical terms, if a manufacturing process were to produce resistors with a true resistance that is normally distributed, we would expect most resistors to have a resistance close to the mean value. In our example, with the mean set at 6 ohms, most resistors will have resistances not too far from 6 ohms. However, due to natural variations in the manufacturing process, some resistors will have slightly higher or lower resistance, and the probability of these can be predicted using the properties of the normal distribution.

For students, it’s crucial to visualize this distribution as a symmetric, bell-shaped curve. The area under the curve represents the cumulative probability, and it's essential to know that about 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations, according to the empirical rule.

Variance and Standard Deviation

Variance and standard deviation are both measures of spread or dispersion in a set of data. Variance is the average of the squared differences from the mean, while the standard deviation is the square root of the variance. These metrics are essential because they provide an indication of how spread out the values of a dataset are.

In the context of our resistor problem, the standard deviation of 0.3 ohm represents the amount of variation in resistance we might expect from the mean of 6 ohms. A smaller standard deviation would indicate that the resistances of the resistors are very consistent, hovering close to the mean value. Conversely, a larger standard deviation would imply more variability and a wider disparity in resistance values.

If the standard deviation is small, the values cluster closely around the mean, resulting in a steeper bell curve. For a larger standard deviation, the values are spread out over a wider range, leading to a flatter bell curve. A strong grasp of these concepts allows students to understand how precision in manufacturing can influence the reliability and performance of electronic components such as resistors.

Z Score

The Z score is a key concept in statistics, as it represents the distance between a data point and the mean, measured in units of standard deviation. It's a way of standardizing different datasets to compare them directly, regardless of their individual scales or units of measurement.

In our resistor example, we calculated a Z score to determine how unusual it would be for the combined resistance of three resistors to exceed 19 ohms. The Z score helps us convert this resistance value into standard deviations away from the expected mean. A high Z score, in absolute terms, signifies that the data point is far from the mean - in the case of positive values, it means the data point is above the mean.

Students should recognize the Z score as a tool to assess the probability of a particular outcome. Specifically, if the Z score of a resistor's resistance is very high, it indicates that such a resistor is not just a typical variation but a significant outlier, which might be of concern in quality control and reliability of electronic circuits.

Cumulative Probability

Cumulative probability is the likelihood that a random variable will take a value less than or equal to a certain number. It is essentially the area under the probability density function (PDF) curve to the left of that number. In the context of the normal distribution curve, it represents the total area under the curve from the far left up to the Z score.

In our exercise, after finding the Z score, we refer to a Z-table to find the cumulative probability associated with that Z score. This cumulative probability represents the likelihood that the total resistance would be less than or equal to our observed value. To find the probability of exceeding a certain value, we subtract the cumulative probability from one.

Understanding cumulative probability is crucial for students, as it helps in making predictions about outcomes based on the distribution of the data. For instance, electronics manufacturers might use cumulative probabilities to ensure that a minimal number of produced resistors fall outside of the expected resistance tolerance, thereby maintaining a high standard of quality control.