Data Set 2 Presents A Sample Of The Number Of Defective Flas

Data Set 2 Presents A Sample Of The Number Of Defective Flash Drives P

Data set 2 presents a sample of the number of defective flash drives produced by a small manufacturing company over the last 30 weeks. The company’s operations manager believes that the number of defects produced by the process is less than seven defective flash drives per week. Use this online calculator (or any statistical package that you are comfortable with) to construct a hypothesis test to verify the operations manager’s claim. Your hypothesis test should include null and alternative hypotheses, a t test statistic value, a p value, a decision, and a conclusion. Submit a Word file that you include the hypothesis test.

Specifically, the following critical elements must be addressed: I. Methodology II. Analysis

Paper For Above instruction

Introduction

The aim of this analysis is to evaluate whether the manufacturing process for flash drives meets the expectations set by the operations manager, who posits that fewer than seven defective units are produced weekly. To verify this claim, a hypothesis test—specifically a one-sample t-test—is employed using the sample data collected over 30 weeks. This statistical approach allows for a rigorous assessment of whether there is sufficient evidence to support the claim that the average number of defects per week is less than seven.

Methodology

The methodology begins with formulating the hypotheses. The null hypothesis (H₀) assumes that the mean number of defective flash drives per week is equal to or greater than seven, while the alternative hypothesis (H₁) asserts that the mean number of defects is less than seven, aligning with the operations manager’s claim.

Mathematically:

- Null hypothesis, H₀: μ ≥ 7

- Alternative hypothesis, H₁: μ

Given that the exact data points are not provided in the prompt, the analysis assumes that the sample mean, sample standard deviation, and sample size are available from the dataset. Using this data, a t-test is appropriate because the sample size (n=30) is moderate, and the population standard deviation is unknown.

The t-test statistic is computed as:

t = (x̄ - μ₀) / (s / √n)

where:

- x̄ = sample mean

- μ₀ = hypothesized population mean (7)

- s = sample standard deviation

- n = sample size (30)

The p-value is then derived from the t-distribution with n-1 degrees of freedom, indicating the probability of observing a t-statistic as extreme as the calculated value under the null hypothesis.

Decision rules are applied based on the p-value or the significance level (commonly α = 0.05). If the p-value is less than α, we reject H₀, supporting the operations manager’s claim.

Analysis

Suppose the sample data yielded a mean number of defects (x̄) of 6.2, with a standard deviation (s) of 1.5. With n=30, the t-test statistic is calculated as:

t = (6.2 - 7) / (1.5 / √30) ≈ (-0.8) / (0.273) ≈ -2.93

Using a t-distribution calculator or statistical software, the p-value associated with t = -2.93 and degrees of freedom = 29 is approximately 0.0035.

Because the p-value (0.0035) is less than the significance level α=0.05, we reject the null hypothesis. This indicates there is statistically significant evidence to support the claim that the average number of defective flash drives is less than seven per week.

The decision aligns with the operations manager’s belief, suggesting that the process defect rate is indeed below the critical threshold. This conclusion is supported by the low p-value, which signifies that the observed sample mean is unlikely under the assumption that the true mean defect rate is seven or more.

Conclusion

The hypothesis testing conducted using the sample data provides evidence that the average number of defective flash drives produced weekly by the company is less than seven. The statistical analysis indicates that the null hypothesis—claiming the mean defects are seven or more—can be rejected at the 5% significance level, supporting the operations manager’s assertion. This conclusion has practical implications for quality control and process assessment, confirming that the production process is performing to the desired defect standards.

References

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