A New Process For Producing Synthetic Diamonds Can Be 803029
A New Process For Producing Synthetic Diamonds Can Be Operated At a Pr
A new process for producing synthetic diamonds can be operated at a profitable level only if the average weight of the diamonds produced by the process is greater than 0.5 karat. To evaluate the profitability of the process, a sample of six diamonds was generated using this new process, with recorded weights 0.46, 0.61, 0.52, 0.48, 0.57, and 0.54 karat. Do the six measurements present sufficient evidence to indicate that the average weight of the diamonds produced by the new process is in excess of 0.5 karat?
Paper For Above instruction
Statistical Evidence for synthetic diamonds exceeding 0.5 karat in weight
In the realm of advanced materials and manufacturing, synthetic diamonds have gained significant attention due to their industrial and commercial applications. The profitability of producing synthetic diamonds hinges crucially on certain critical parameters, one of which is the average weight of the diamonds produced. Specifically, for a new synthetic diamond production process to be considered economically viable, the average diamond weight must exceed 0.5 karat. This paper investigates whether the current sample provides sufficient statistical evidence to support this claim.
Introduction
The production of synthetic diamonds involves complex processes, where the economic feasibility depends heavily on the quality and size of the diamonds generated. To maintain profitability, manufacturers often set threshold parameters; in this case, a minimum average weight of 0.5 karat for the diamonds produced. When evaluating whether a new process meets this criterion, statistical hypothesis testing becomes a valuable tool. The current study analyzes a small sample of six diamonds, with weights: 0.46, 0.61, 0.52, 0.48, 0.57, and 0.54 karat, to assess if the true population mean exceeds 0.5 karat.
Methodology
The statistical approach employed here is a one-sample t-test, appropriate due to the small sample size (n=6) and the unknown population standard deviation. The null hypothesis (H₀) states that the population mean weight μ is 0.5 karat (H₀: μ = 0.5), while the alternative hypothesis (H₁) posits that the mean exceeds 0.5 karat (H₁: μ > 0.5). This is a one-tailed test, designed to determine whether there is sufficient evidence to support the claim that the process produces, on average, diamonds larger than 0.5 karat.
Calculations involve determining the sample mean (\(\bar{x}\)), sample standard deviation (s), and then computing the t-statistic:
\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\]
where \(\mu_0 = 0.5\), and n=6.
Results
First, compute the sample mean:
\[
\bar{x} = \frac{0.46 + 0.61 + 0.52 + 0.48 + 0.57 + 0.54}{6} = \frac{3.18}{6} = 0.53 \text{ karat}
\]
Next, calculate the sample standard deviation:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]
Calculating each squared deviation:
\[
(0.46 - 0.53)^2 = 0.0049
\]
\[
(0.61 - 0.53)^2 = 0.0064
\]
\[
(0.52 - 0.53)^2 = 0.0001
\]
\[
(0.48 - 0.53)^2 = 0.0025
\]
\[
(0.57 - 0.53)^2 = 0.0016
\]
\[
(0.54 - 0.53)^2 = 0.0001
\]
Sum of squared deviations:
\[
0.0049 + 0.0064 + 0.0001 + 0.0025 + 0.0016 + 0.0001 = 0.0156
\]
Standard deviation:
\[
s = \sqrt{\frac{0.0156}{5}} = \sqrt{0.00312} \approx 0.0559
\]
Determine the t-statistic:
\[
t = \frac{0.53 - 0.5}{0.0559 / \sqrt{6}} = \frac{0.03}{0.0559 / 2.45} = \frac{0.03}{0.0228} \approx 1.32
\]
With degrees of freedom df = 5, referencing a t-distribution table for a significance level of 0.05 (or 95% confidence), the critical t-value for a one-tailed test is approximately 2.015.
Discussion
The computed t-statistic of approximately 1.32 is less than the critical value of 2.015. Therefore, we do not have sufficient evidence at the 5% significance level to reject the null hypothesis that the mean weight is equal to 0.5 karat. The sample mean of 0.53 karat, while slightly above 0.5, does not statistically support that the true average exceeds 0.5 with high confidence.
This outcome suggests that, although the average observed weight is above 0.5, due to variability and the small sample size, there isn't enough evidence to claim the process produces diamonds larger than 0.5 karat on average. Increasing sample size or reducing variability could provide more definitive results.
Conclusion
Based on the analysis, the current sample does not provide sufficient statistical evidence to conclude that the average weight of diamonds produced by the new process exceeds 0.5 karat at the 5% significance level. This indicates that further testing, with larger samples or improved process control, is necessary to confidently establish the profitability threshold being met.
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