Statistical Analysis Of Copper Wire Breaking Strength A Hypothesis Test

In the realm of materials science and engineering, understanding the mechanical properties of materials is paramount. Breaking strength, a critical parameter, dictates a material's ability to withstand tensile stress before fracturing. This article delves into a statistical analysis of the breaking strength of copper wires, a ubiquitous material in electrical applications. We embark on a journey to test a specific hypothesis: whether the mean breaking strength of a large lot of copper wires can be confidently assumed to be 578 kg weight. This exploration will involve employing statistical tools and techniques to scrutinize a sample of copper wires, ultimately shedding light on the reliability and consistency of this material in practical applications. Through a meticulous examination of the provided data, we aim to unravel the intricacies of copper wire strength, providing valuable insights for engineers, scientists, and anyone with an interest in the behavior of materials under stress.

To begin our analysis, let's first present the data at hand. We have a sample of ten copper wires drawn from a large lot. The breaking strengths of these wires, measured in kilograms (kg) weight, are as follows:

578, 572, 570, 568, 572, 571, 570, 572, 596, 548

This dataset forms the foundation of our statistical investigation. Each data point represents the breaking strength of an individual copper wire, providing a snapshot of the material's performance under tensile stress. Our goal is to leverage this information to make inferences about the entire lot of copper wires from which this sample was drawn. By employing statistical techniques, we can move beyond the individual data points and gain a broader understanding of the material's overall strength characteristics. This data presentation serves as the springboard for our analysis, setting the stage for a deeper exploration of the breaking strength of copper wires.

At the heart of our statistical investigation lies a specific question: Can the mean breaking strength of the entire lot of copper wires be assumed to be 578 kg weight? To address this question, we formulate a hypothesis, a statement about the population parameter that we aim to test. In statistical hypothesis testing, we typically frame two competing hypotheses: the null hypothesis and the alternative hypothesis.

• Null Hypothesis (H0): The mean breaking strength of the lot is 578 kg weight.
• Alternative Hypothesis (H1): The mean breaking strength of the lot is not 578 kg weight.

The null hypothesis represents the status quo, the statement we assume to be true unless sufficient evidence suggests otherwise. In our case, the null hypothesis posits that the mean breaking strength of the copper wires is indeed 578 kg weight. The alternative hypothesis, on the other hand, challenges the null hypothesis, proposing that the mean breaking strength is different from 578 kg weight. This difference could be either higher or lower, making our alternative hypothesis a two-tailed test.

By formulating these hypotheses, we establish a framework for our statistical analysis. We will use the sample data to gather evidence and determine whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis. This process allows us to make informed decisions about the true mean breaking strength of the copper wires, based on the available data.

To rigorously test our hypothesis, we must select an appropriate statistical test. Given our objective of comparing the sample mean to a hypothesized population mean, and considering that the population standard deviation is unknown, the t-test emerges as the ideal choice. The t-test is a powerful statistical tool designed specifically for situations like ours, where we have a sample of data and want to draw inferences about the population mean. It takes into account the sample size, the sample mean, the sample standard deviation, and the hypothesized population mean to calculate a test statistic, which we can then use to assess the evidence against the null hypothesis.

There are different types of t-tests, but for our scenario, the one-sample t-test is most suitable. This test is specifically designed for situations where we have a single sample and want to compare its mean to a known or hypothesized population mean. In our case, we have a single sample of copper wire breaking strengths, and we want to compare its mean to the hypothesized population mean of 578 kg weight. The one-sample t-test will allow us to determine whether the difference between the sample mean and the hypothesized population mean is statistically significant, or whether it is likely due to random chance.

The t-test operates under certain assumptions, such as the data being normally distributed. We will need to verify these assumptions before proceeding with the test. However, the t-test is generally robust to violations of normality, especially with larger sample sizes. By carefully selecting the t-test, we ensure that our statistical analysis is grounded in sound principles, providing a reliable basis for drawing conclusions about the breaking strength of copper wires.

Now, let's delve into the data analysis and calculations required to perform the t-test. We will first compute the sample mean and sample standard deviation, essential ingredients for the t-test formula. The sample mean, denoted as x̄, represents the average breaking strength of the copper wires in our sample. It is calculated by summing all the breaking strengths and dividing by the number of observations.

Sample Mean (x̄):

x̄ = (578 + 572 + 570 + 568 + 572 + 571 + 570 + 572 + 596 + 548) / 10 = 570.9 kg weight

Next, we calculate the sample standard deviation, denoted as s, which measures the spread or variability of the data around the sample mean. A higher standard deviation indicates greater variability, while a lower standard deviation suggests that the data points are clustered more closely around the mean.

Sample Standard Deviation (s):

To calculate the sample standard deviation, we first find the deviations of each data point from the sample mean, square these deviations, sum the squared deviations, divide by the number of observations minus 1 (n-1), and finally take the square root of the result. This process can be represented by the following formula:

s = √[∑(xi - x̄)² / (n - 1)]

Where:

• xi represents each individual data point
• x̄ is the sample mean
• n is the sample size

After performing the calculations, we find the sample standard deviation to be:

s ≈ 13.57 kg weight

With the sample mean and sample standard deviation in hand, we can now calculate the t-statistic, the cornerstone of the t-test. The t-statistic measures the difference between the sample mean and the hypothesized population mean, relative to the variability within the sample. It is calculated using the following formula:

T-statistic (t):

t = (x̄ - μ) / (s / √n)

Where:

• x̄ is the sample mean
• μ is the hypothesized population mean (578 kg weight)
• s is the sample standard deviation
• n is the sample size

Plugging in the values, we get:

t = (570.9 - 578) / (13.57 / √10) ≈ -1.65

The t-statistic, -1.65, quantifies the discrepancy between our sample data and the null hypothesis. The negative sign indicates that the sample mean is lower than the hypothesized population mean. To determine the significance of this discrepancy, we need to compare the t-statistic to a critical value or calculate a p-value.

The p-value is a crucial concept in hypothesis testing. It represents the probability of observing a test statistic as extreme as, or more extreme than, the one we calculated, assuming that the null hypothesis is true. In simpler terms, it tells us how likely it is to see our sample data if the true population mean is indeed 578 kg weight. A small p-value suggests that our observed data is unlikely under the null hypothesis, providing evidence against it.

To determine the p-value, we need to consider the t-distribution and the degrees of freedom. The degrees of freedom (df) represent the number of independent pieces of information available to estimate a parameter. In our case, for a one-sample t-test, the degrees of freedom are calculated as:

df = n - 1 = 10 - 1 = 9

With 9 degrees of freedom, we can consult a t-distribution table or use statistical software to find the p-value associated with our t-statistic of -1.65. Since our alternative hypothesis is two-tailed (the mean breaking strength is not 578 kg weight), we need to consider both tails of the t-distribution.

Using a t-distribution table or statistical software, we find that the p-value for a two-tailed t-test with t = -1.65 and df = 9 is approximately:

p-value ≈ 0.134

This p-value of 0.134 indicates that there is a 13.4% chance of observing a sample mean as far away from 578 kg weight as our sample mean (570.9 kg weight), if the true population mean is indeed 578 kg weight. Now, we need to compare this p-value to our chosen significance level to make a decision about our hypothesis.

To draw a conclusion from our statistical analysis, we need to compare the p-value to a predetermined significance level, often denoted as α (alpha). The significance level represents the threshold for rejecting the null hypothesis. Common significance levels are 0.05 (5%) and 0.01 (1%). If the p-value is less than or equal to the significance level, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.

In our case, let's assume a significance level of α = 0.05. This means that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis when it is actually true (a Type I error).

Comparing our p-value (0.134) to the significance level (0.05), we observe that:

p-value (0.134) > α (0.05)

Since the p-value is greater than the significance level, we fail to reject the null hypothesis. This means that we do not have sufficient evidence to conclude that the mean breaking strength of the lot of copper wires is significantly different from 578 kg weight.

In conclusion, based on our analysis of the sample data, we cannot reject the hypothesis that the mean breaking strength of the copper wires is 578 kg weight. This suggests that the manufacturer's claim of a mean breaking strength of 578 kg weight is plausible, given the available evidence.

It is important to note that failing to reject the null hypothesis does not necessarily mean that the null hypothesis is true. It simply means that we do not have enough evidence to reject it. There is still a possibility that the true mean breaking strength is different from 578 kg weight, but our sample data does not provide strong enough evidence to support this claim. Further investigation with a larger sample size or other statistical methods might be necessary to draw more definitive conclusions.

As with any statistical analysis, it's crucial to acknowledge potential sources of error that could influence our results. Understanding these limitations allows for a more nuanced interpretation of our findings and guides future investigations. Several factors could contribute to discrepancies between our sample data and the true population characteristics.

• Sampling Error: Our analysis is based on a sample of ten copper wires, which is a small subset of the entire lot. Sampling error arises from the fact that a sample may not perfectly represent the population from which it is drawn. By chance, our sample might contain wires that are slightly stronger or weaker than the average for the entire lot. Increasing the sample size can help reduce sampling error.
• Measurement Error: The breaking strength measurements themselves could be subject to error. The testing equipment might have limitations in its precision, or there might be variations in how the tests were conducted. Careful calibration of equipment and standardized testing procedures are essential to minimize measurement error.
• Non-Normality: The t-test assumes that the data is normally distributed. While the t-test is generally robust to violations of normality, particularly with larger sample sizes, significant departures from normality could affect the accuracy of the p-value. We did not formally test for normality in this analysis, which is a potential limitation. Visual inspection of the data (e.g., using a histogram or normal probability plot) or formal normality tests could help assess this assumption.
• Outliers: The presence of outliers, extreme values that deviate significantly from the rest of the data, can disproportionately influence the sample mean and standard deviation. In our data, the breaking strength of 596 kg weight is noticeably higher than the other values, which could be considered a potential outlier. While we did not remove this data point, further analysis could explore the impact of outliers on the results.

By acknowledging these potential sources of error, we provide a more comprehensive understanding of the limitations of our analysis. This transparency enhances the credibility of our conclusions and highlights areas where further investigation might be beneficial.

In summary, our statistical analysis of the breaking strength of copper wires, using a one-sample t-test, did not provide sufficient evidence to reject the null hypothesis that the mean breaking strength of the lot is 578 kg weight. While this suggests that the manufacturer's claim is plausible, it's important to interpret this finding with caution, considering the potential sources of error discussed earlier.

Based on our analysis, we recommend the following:

• Increase Sample Size: A larger sample size would provide a more representative picture of the population and reduce the impact of sampling error. We recommend testing a larger number of copper wires to obtain more precise estimates of the mean breaking strength.
• Assess Normality: Formally test the assumption of normality. While the t-test is robust to minor deviations from normality, it's prudent to confirm that the data is reasonably normally distributed. Techniques like the Shapiro-Wilk test or visual inspection of histograms and normal probability plots can be used.
• Investigate Outliers: Further investigate potential outliers. Determine whether they are genuine data points or the result of measurement errors or other anomalies. Consider the impact of outliers on the results, potentially by performing the t-test with and without the outlier(s).
• Control for Measurement Error: Implement rigorous quality control procedures during the breaking strength testing process. This includes calibrating testing equipment regularly, using standardized testing protocols, and training personnel to minimize measurement error.
• Consider Alternative Tests: If the normality assumption is severely violated, consider non-parametric tests, such as the Wilcoxon signed-rank test, which do not rely on the assumption of normality.

By implementing these recommendations, future investigations can provide a more robust and reliable assessment of the breaking strength of copper wires. This will ultimately contribute to ensuring the quality and reliability of electrical applications that rely on this essential material.