New Process For Producing Synthetic Diamond

Question

New Process For Producing Synthetic Diamon Due Wednesday at noon 1200 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? To answer this question conduct an appropriate test of hypothesis using the five step process outlined in our textbook and utilized in the solutions to the Chapter 8 review problems, which can be accessed via the link in the Assignments, Tests & Quizzes sub-section in the weekly lesson that can be found in the Lessons section of our classroom. Keep in mind that your post must be made by 11:55PM EASTERN time on Wednesday during the week in which a discussion question is posed. I will evaluate your responses to each of these questions using a 0 to 10 point scale, and your contribution to each of the Discussion Forums will count as 1.25 percent of the overall course grade for a total of 10 percent. My evaluation of your post will be based on the extent to which you participated and fostered a positive and effective learning environment--for yourself and others. Participating and sharing are the keys. Naturally, simply copying someone else's post is prohibited.

Dr. Jack HW Helper · Accepted Answer

Introduction The production of synthetic diamonds has become an increasingly profitable industry, driven by advancements in material science and manufacturing techniques. An essential aspect of maintaining profitability is verifying whether new production processes yield diamonds of sufficient weight—specifically, greater than 0.5 karats. To assess this, a hypothesis testing approach is necessary, using sample data to make inferences about the true mean weight of diamonds produced by the process. Methodology The core of the investigation involves conducting a one-sample t-test for the mean. The null hypothesis ($H_0$) posits that the mean weight of the diamonds ($\mu$) is less than or equal to 0.5 karats, while the alternative hypothesis ($H_a$) suggests that $\mu$ exceeds 0.5 karats. Mathematically: - $H_0: \mu \leq 0.5$ - $H_a: \mu > 0.5$ The sample data consists of six diameters: 0.46, 0.61, 0.52, 0.48, 0.57, and 0.54 karats. This data will be used to calculate the sample mean and standard deviation, which serve as inputs for the t-test. The justification for using a t-test hinges on the small sample size and the assumption that the weights are approximately normally distributed. Since the sample size is small (n=6), the t-distribution provides an appropriate framework for statistical inference. Calculation and Analysis Calculating the sample mean: $$ \bar{x} = \frac{0.46 + 0.61 + 0.52 + 0.48 + 0.57 + 0.54}{6} = \frac{3.18}{6} = 0.53 \text{ karats} $$ Calculating the sample standard deviation: $$ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} $$ where, $$ \sum (x_i - \bar{x})^2 = (0.46-0.53)^2 + (0.61-0.53)^2 + (0.52-0.53)^2 + (0.48-0.53)^2 + (0.57-0.53)^2 + (0.54-0.53)^2 $$ $$ = (-0.07)^2 + (0.08)^2 + (-0.01)^2 + (-0.05)^2 + (0.04)^2 + (0.01)^2 $$ $$ = 0.0049 + 0.0064 + 0.0001 + 0.0025 + 0.0016 + 0.0001 = 0.0156 $$ $$ s = \sqrt{\frac{0.0156}{5}} = \sqrt{0.00312} \approx 0.0559 $$ Next, compute the t-statistic: \[ t = \frac{\bar{x} - \mu_0}{s / \sqr

New Process For Producing Synthetic Diamond

New Process For Producing Synthetic Diamon

Paper For Above instruction

Introduction

Methodology

Calculation and Analysis

Discussion of Rationale

Conclusion

References