91Ó°ÊÓ

Electronic circuit The design of an electronic circuit for a toaster calls for a 100-ohm resistor and a \(250-\) ohm resistor connected in series so that their resistances add. The components used are not perfectly uniform, so that the actual resistances vary independently according to Normal distributions. The resistance of 100 -ohm resistors has mean 100 ohms and standard deviation 2.5 ohms, while that of 250 -ohm resistors has mean 250 ohms and standard deviation 2.8 ohms. (a) What is the distribution of the total resistance of the two components in series for a randomly selected toaster? (b) Find the probability that the total resistance for a randomly selected toaster lies between 345 and 355 ohms.

Step by step solution

01

Understand the Distribution of Total Resistance

The problem states that the resistances of the two resistors follow normal distributions: the 100-ohm resistor has mean \(\mu_1 = 100\) ohms and standard deviation \(\sigma_1 = 2.5\) ohms, while the 250-ohm resistor has mean \(\mu_2 = 250\) ohms and standard deviation \(\sigma_2 = 2.8\) ohms. Because resistances in series add, the total resistance \(R\) is the sum of the two random variables. If \(X\) and \(Y\) are normally distributed random variables with means \(\mu_1\) and \(\mu_2\) and standard deviations \(\sigma_1\) and \(\sigma_2\), then \(Z = X + Y\) is also normally distributed with mean \(\mu = \mu_1 + \mu_2\) and variance \(\sigma^2 = \sigma_1^2 + \sigma_2^2\).

02

Calculate the Mean and Variance of Total Resistance

Compute the mean of the total resistance: \(\mu = 100 + 250 = 350\) ohms. Next, calculate the variance of the total resistance: \(\sigma^2 = 2.5^2 + 2.8^2 = 6.25 + 7.84 = 14.09\). The standard deviation \(\sigma\) is the square root of the variance: \(\sigma = \sqrt{14.09} \approx 3.75\) ohms.

03

Determine the Distribution of the Total Resistance

The distribution of the total resistance \(R\) is Normal with mean \(\mu = 350\) ohms and standard deviation \(\sigma = 3.75\) ohms. Therefore, \(R \sim N(350, 3.75^2)\).

04

Find Probability Using the Normal Distribution

We need to find the probability that the total resistance \(R\) lies between 345 and 355 ohms. Standardize these using the normal distribution formula: \(Z = \frac{X - \mu}{\sigma}\). The standardized values are \(Z_1 = \frac{345 - 350}{3.75} \approx -1.33\) and \(Z_2 = \frac{355 - 350}{3.75} \approx 1.33\).

05

Use Z-tables to Find Probability

Consult the Z-table to find probabilities. For \(Z_1 = -1.33\), the probability that \(Z \leq -1.33\) is approximately 0.0918. For \(Z_2 = 1.33\), the probability that \(Z \leq 1.33\) is approximately 0.9082. The probability \(P(345 < R < 355)\) is \(P(Z_1 < Z < Z_2) = P(Z < 1.33) - P(Z < -1.33) = 0.9082 - 0.0918 = 0.8164\).

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Key Concepts

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

Probability Calculation

Probability calculation is a way to quantify the likelihood that a certain event will happen. It helps us understand how often we can expect a particular outcome.
In the context of the exercise, probability calculation is used to determine how likely it is that the resistance of the two resistors in series will fall within a specific range. When we calculate probabilities associated with normally distributed data, we use the properties of a normal distribution. For instance:

• To convert an actual range into a probability, we "standardize" it using a formula.
• This standardization involves finding a Z-score, which tells us how many standard deviations away a value is from the mean.

Understanding the standard normal distribution and using the Z-scores allows us to use statistical tables (Z-tables) to find the probability that a resistance will be within a particular range.

Resistor Resistance

Resistor resistance is a measure of how much a resistor opposes the flow of electric current. In this exercise, we are analyzing two resistors connected in series.
When resistors are in series, their individual resistances add up to give the total resistance. Several key points to consider about resistor resistance:

• The 100-ohm and 250-ohm resistors are not perfect; their actual resistances vary slightly due to manufacturing differences.
• This variation is modeled using normal distributions, which allows us to make predictions about possible resistance values.

The resistances vary independently, meaning one resistor’s resistance does not impact the other’s variation. This independence is crucial when performing calculations using the normal distribution.

Random Variables

Random variables are used to represent numerical outcomes of random phenomena. When dealing with random variables, we're interested in their distributions, which tell us the probability of each possible outcome.
For the resistors, each can be thought of as a random variable due to its normally distributed resistance. Some important facts about random variables:

• When you add two independent normally distributed random variables, the result is also normally distributed.
• The mean of the resulting variable is the sum of the means of the individual variables.
• The variance of the resulting variable is the sum of the variances of the individual variables. This is because they vary independently.

In this case, the total resistance is a new random variable with a calculated mean and variance based on the original resistances.

Statistical Analysis

Statistical analysis involves collecting and examining data to find patterns or trends. It requires us to use mathematical methods to better understand randomness and make informed decisions based on this understanding.
In this exercise, statistical analysis is employed to determine the probability of different outcomes regarding resistance. Key aspects of statistical analysis applied here include:

• The use of normal distribution to model real-world resistance variability.
• Calculating and using Z-scores to find probabilities within this variability model.
• Interpreting standard statistical tables (like the Z-table) to understand these probabilities in a meaningful way.

Statistical analysis helps bridge the gap between theoretical models, like the normal distribution, and practical applications, such as ensuring electronic components meet required resistance specifications.