A New Process For Producing Synthetic Diamonds Can Be Operat
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? 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 are posted in the Review Problem Sets Solutions folder in the Resources section of our classroom.
Paper For Above instruction
Introduction
The evaluation of the new synthetic diamond production process hinges on statistical hypothesis testing to determine whether the mean weight of the diamonds exceeds 0.5 karat. Given the importance of profitability and industrial viability, it is essential to statistically verify this claim based on sample data. In this analysis, we implement a formal hypothesis test, adhering to the five-step process common in inferential statistics, to assess if the evidence from the sample weights suggests an average weight greater than 0.5 karat.
Step 1: State the Hypotheses
The null hypothesis (H0) assumes that the average weight of diamonds produced by the process is equal to 0.5 karat, formally:
H0: μ = 0.5
The alternative hypothesis (Ha) posits that the average weight exceeds 0.5 karat:
Ha: μ > 0.5
This is a one-tailed test designed to determine whether the process produces, on average, heavier diamonds than the threshold value critical for profitability.
Step 2: Select the Significance Level
The significance level (α) is set at 0.05, corresponding to a 5% risk of incorrectly rejecting the null hypothesis when it is true. This conventional threshold balances the risks of Type I and Type II errors in industrial decision-making contexts.
Step 3: Collect Data and Calculate the Test Statistic
The sample weights are: 0.46, 0.61, 0.52, 0.48, 0.57, and 0.54 karat.
First, compute the sample mean:
\[
\bar{x} = \frac{0.46 + 0.61 + 0.52 + 0.48 + 0.57 + 0.54}{6} = \frac{3.18}{6} = 0.53
\]
Next, calculate the sample standard deviation (s):
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]
Calculating deviations:
\[
(0.46 - 0.53)^2 = 0.0049 \\
(0.61 - 0.53)^2 = 0.0064 \\
(0.52 - 0.53)^2 = 0.0001 \\
(0.48 - 0.53)^2 = 0.0025 \\
(0.57 - 0.53)^2 = 0.0016 \\
(0.54 - 0.53)^2 = 0.0001
\]
Sum of squared deviations:
\[
0.0049 + 0.0064 + 0.0001 + 0.0025 + 0.0016 + 0.0001 = 0.0166
\]
Sample standard deviation:
\[
s = \sqrt{\frac{0.0166}{5}} \approx \sqrt{0.00332} \approx 0.0576
\]
The standard error:
\[
SE = \frac{s}{\sqrt{n}} = \frac{0.0576}{\sqrt{6}} \approx \frac{0.0576}{2.45} \approx 0.0235
\]
The test statistic (t):
\[
t = \frac{\bar{x} - \mu_0}{SE} = \frac{0.53 - 0.5}{0.0235} \approx 1.28
\]
Note: Since the sample size is small, a t-test is appropriate.
Step 4: Make a Decision Using the p-value
Using a t-distribution table or calculator with degrees of freedom df = 5, the p-value for t = 1.28 in a one-tailed test is approximately 0.132.
Because p > 0.05, we fail to reject the null hypothesis. There is insufficient evidence at the 5% significance level to conclude that the average weight exceeds 0.5 karat.
Step 5: Draw a Conclusion
Based on the data and the hypothesis test, we conclude that there is not enough statistical evidence to suggest that the process produces diamonds with an average weight greater than 0.5 karat. The observed mean of 0.53 karat does not significantly exceed the threshold at the 5% significance level, indicating that further data or increased sample size may be necessary to definitively assess profitability.
Additional Context: Hypotheses for the Diamond Weight Process
The core statistical decision aligns with the practical question of whether the process meets the profitability threshold. Since the evidence does not confirm that the mean weight exceeds 0.5 karat, it suggests that the process may need modification or further validation, especially considering the critical economic implications.
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