Hypothesis Testing: The Lazer Company Has A Contract To Prod

hypothesis Testingthe Lazer Company Has A Contract To Produce A Part

Hypothesis testing is a fundamental statistical method used to determine whether there is enough evidence to support a specific claim or hypothesis about a population parameter based on sample data. In the context of the Lazer Company, which produces parts for Boeing Corporation, hypothesis testing can be employed to assess if their production process meets the specified mean diameter requirement of 6 inches, given that the process already controls the standard deviation effectively.

The company's goal is to verify whether the most recent production batch adheres to the mean diameter specification of 6 inches. This involves formulating appropriate hypotheses, collecting sample data, and applying statistical tests to inform decision-making. By doing so, Lazer can ensure compliance with contractual quality standards and maintain their reputation with Boeing.

Paper For Above instruction

To evaluate whether the production process of Lazer Company is meeting the specified mean diameter of 6 inches, hypothesis testing provides a systematic approach. The initial step involves establishing the null and alternative hypotheses that reflect the company's claim versus the counterclaim based on observed data.

Hypotheses Formulation

The null hypothesis (H0) is a statement asserting that the true mean diameter of the parts produced is equal to the specification, which is 6 inches. Mathematically, this is represented as:

H0: μ = 6 inches

The alternative hypothesis (H1) represents the condition that the process does not meet the specification, indicating that the mean diameter differs from 6 inches. Since the concern is whether the process deviates in either direction, a two-tailed test is appropriate:

H1: μ ≠ 6 inches

Given that the process already controls the standard deviation at 0.10 inch, the decision to use a Z-test is justified because the sample size (n=200) is sufficiently large for the Central Limit Theorem to apply, which normalizes the sampling distribution of the sample mean.

Decision Rule Development

Assuming a significance level (α) of 0.01, the critical z-values for a two-tailed test can be obtained from standard normal distribution tables. The critical values are approximately ±2.576, meaning any test statistic outside this range would lead to rejecting the null hypothesis.

The test statistic (Z) is computed using the formula:

Z = (X̄ - μ0) / (σ / √n)

Where:

  • X̄ is the sample mean
  • μ0 is the hypothesized population mean (6 inches)
  • σ is the population standard deviation (0.10 inch)
  • n is the sample size (200)

With the sample size of 200 parts, the standard error (SE) of the mean is:

SE = σ / √n = 0.10 / √200 ≈ 0.00707

Using the sample mean of 6.03 inches, the calculated Z-value is:

Z = (6.03 - 6) / 0.00707 ≈ 4.24

This Z-value exceeds the critical value of 2.576, indicating that the sample mean significantly differs from the target mean at the 1% significance level.

Conclusion and Recommendations

Based on the Z-test analysis, the sample mean of 6.03 inches leads to rejecting the null hypothesis that the process produces parts with a mean diameter of exactly 6 inches. The evidence suggests that the process's mean has shifted and no longer meets the contractual requirements.

Therefore, Lazer Company should conclude that the current process is not compliant with the required mean diameter of 6 inches. Continuing with the current process could jeopardize contractual obligations with Boeing, potentially resulting in penalties or loss of business.

It is recommended that Lazer investigate the causes of this shift, possibly adjusting their process parameters, enhancing quality control measures, or implementing process improvements. Further sampling and ongoing monitoring should be undertaken to verify the effectiveness of these interventions and ensure future compliance.

References

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