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# Resistance of a Resistor: Blame the Students or Blame the Resistors?

## Abstract

The purpose of this study is to find the proportion of defective resistors within the arsenal of resistors that Midwood High School uses for labs.

## Inspiration

This incentive was inspired by typical lab work from the physical science departments in which students were seldom blamed for inconsistent data collected by using possibly faulty resistors. Our goal is to determine whether the students were at fault when using resistors or the resistors themselves.

## Introduction

Resistors are devices that are used in various technological devices in order to control current flow and therefore making the product efficient. They can be found in any electronic equipment that uses a circuit in order to control its functions. The strength or resistance of a resistor is measured in Ohms (Ω). Resistances of standard resistors can be found by using a standardized color code. Under this code, resistors are imprinted with four bands of colors. The code is shown below.

Resistor Color Code
first & second bands
(first & second digits)
third band
(multiplier)
fourth band
(tolerance)
black   0   black   1   none   ±20%
brown   1   brown   10   silver   ±10%
red   2   red   100   gold   ±5%
orange   3   orange   1,000
yellow   4   yellow   10,000
green   5   green   100,000
blue   6   blue   1,000,000
violet   7   silver   0.01
gray   8   gold   0.1
white   9

To find the resistance, the numbers associated with the first and second codes is first written next to each other. Then the multiplier, determined by the third color is multiplied to the number formed by the first two. The fourth color is the tolerance of the resistor. Since the resistance changes in different environments, it is essential to have a range of values, which are acceptable. Tolerance tells the acceptable variation of the resistor's resistance from the value predicted by the first three colors.

## Procedure

1. First we did a simple random sample of 99 resistors from the population of all resistors at Midwood with a gold tolerance. This was done by labeling each resistor with a unique number and picking 99 of these, using a random number table. The fact that a simple random sample is done, will become critical later on.
2. We connected the multimeter and set it to the appropriate mode for measuring resistance.
3. Then, one by one, we connected each resistor to the multimeter and recorded the resistance of each resistor. Additionally, we recorded the color code.

A diagram of the procedure is shown below.

## Analysis

Since it was impractical to study all the resistors in Midwood, we first studied a simple random sample instead and predicted a range of population values based on the sample results. Then we limited our study to resistors with gold tolerance, therefore, effectively eliminating a compounding factor.

Any resistor with a "% deviation" greater than the uniform sample tolerance level (5%) was categorized as a defective. In the simple random sample that we conducted, 3 of the 99 tested resistors were defective. Therefore, 96 of the 99 were not defective.

To process the raw data, we first calculated the predicted resistance based on the color code. Then we found the "% deviation" of each resistor by using the formula below.

% Deviation = (Calculated Resistance − Predicted Resistance)/(Predicted Resistance)

The data from our study can be found here.

### Descriptive Analysis

The table below contains the numerical summary of statistics for the variable "% deviation." It should be noted that all the summary numbers except the maximum and the range fall below the 5% cutoff. Additionally, it should be noted that the sample size is greater than 40.

Total Count Mean Standard Deviation Minimum Q1 Median Q2 Maximum Range IQR
99 3.41 1.12 1.1 2.7 3.2 4.255 8.1818 7.0818 1.555

The picture below is the histogram for the variable "% deviation." The normal curve has also been drawn in to show that central limits theorem indeed applies. Meaning, even though the population might not be normally distributed, any sample size greater than 40 and less than 10% of the population will be. This assumption is necessary for the statistical test done later.

The picture below is the box plot for the variable "% deviation." It should be noted that there are two outliers. However, unlike most studies, these outliers were not removed from the data because these were two of the very few defective resistors in the sample.

### 1-Proportion Z-Test

Using this test, we found a 95% confidence interval for the population proportion of defective resistors in Midwood High School.

#### Hypothesis:

Null Hypothesis: p0 (proportion of defective resistors) = 0

Alternative Hypothesis: pa (proportion of defective resistors) > 0

#### Conditions:

1. Simple Random Sample - Fulfilled (Procedure)
2. Sample Size > 40 - Fulfilled (Descriptive Analysis)
3. Population > 10n - It is logical to assume that there are more than 990 resistors in Midwood high school.
4. npa>10 and n(1-pa)>10 - Since npa = (.0303)*(99) = 3, this condition is not fulfilled. However, the purpose of this test is to ensure that the sample size is large enough and clearly a sample size of 99 is large enough. Therefore, we proceeded with the test.

#### Test Statistics:

Since the histogram clearly shows that the population is normally distributed, we found the z-score by using a normally distributed curve.

Where

0.0303 ± (1.6449)*√((3/99)(96/99))/99
0.0303 ± 0.02834
0.00196 - 0.05864

This result is graphically shown below in a normally distributed probability model for defective resistors. The gray region shows likely values for population proportion and the black region shows highly unlikely values.

## Conclusion

Based on the study, since the the 95% confidence interval in greater than 0%, we concluded that some proportion of resistors in Midwood High School are indeed defective. The exact proportion of defective resistors in Midwood High School is somewhere between 0.196% and 5.864%.

It should be noted that the proportion of defective resistors is really small. Additionally, once we analyzed the data carefully, we noted that only resistors with small resistances gave us a "% deviation" greater than 5%.

Therefore, we concluded that students are indeed not at fault if they get bad results in the lab, given that they followed the procedure correctly. Additionally, since only the resistors with small resistances appeared defective, they can avoid defective resistors by using resistors with high resistance.

## Sources of Error

1. The resistors have been used constantly for lab work, and have been exposed to regular wear and tear. That could have altered the values.
2. It was impractical to collect all the resistors in Midwood High School. Therefore, we did a simple random sample of a convenient sample, which was provided to us by a teacher. If the sample is not representative of the population, that would invalidate the test.

Anurag Panda, Raj Mathew, Suzanna Zhou -- 2007

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