Hypothesis Testing Is Used In Business To Test Assumptions

Hypothesis Testing Is Used In Business To Test Assumptions And Theorie

Hypothesis testing is used in business to evaluate assumptions and theories by analyzing observed data to determine whether a presumed statement about a population parameter is supported by evidence. In this scenario, a production manager is testing whether the average weight of packaged grass seed bags meets the expected 50 pounds.

The null hypothesis (Ho) states that the true mean weight of the bags is 50 pounds, reflecting that the machinery is correctly calibrated. The alternative hypothesis (H1) suggests that the mean weight is less than 50 pounds, indicating potential underfilling by the machinery. Mathematically:

- Ho: μ = 50

- H1: μ

Using a significance level (α) of 0.05, we determine the critical value from the t-distribution (assuming the population standard deviation is unknown and the sample size is small). For a one-tailed test at α=0.05 with degrees of freedom (df) of n-1=19, the critical t-value is approximately -1.729 (from t-tables).

The decision rule is: if the calculated test statistic is less than the critical value (-1.729), reject Ho; otherwise, do not reject Ho.

To compute the test statistic, first calculate the sample mean and sample standard deviation. Using the measured weights:

45.6, 49.5, 47.7, 46.7, 47.6, 48.8, 50.5, 48.6, 50.2, 51.5, 46.9, 50.2, 47.8, 49.9, 49.3, 49.8, 53.1, 49.3, 49.5, 50.1

The sample mean (x̄) is approximately 49.12 pounds, and the sample standard deviation (s) is approximately 2.15 pounds.

The test statistic (t) is calculated as:

t = (x̄ - μ₀) / (s / √n)

t = (49.12 - 50) / (2.15 / √20) ≈ -0.88 / (2.15 / 4.472) ≈ -0.88 / 0.48 ≈ -1.83

Since the calculated t-value (-1.83) is less than the critical value (-1.729), we reject the null hypothesis.

Conclusion: The evidence suggests that the average weight of the bags is less than 50 pounds, indicating the bags are likely underfilled. It would be prudent for machinery to be recalibrated to ensure proper packaging and compliance with standards.

Paper For Above instruction

In the realm of business quality control, hypothesis testing serves as a vital statistical tool to validate assumptions about production processes and product specifications. The scenario involving the underfilled grass seed bags exemplifies how statistical inference can be applied to real-world manufacturing concerns — specifically, testing whether machinery maintains the correct packaging weight.

The foundational step in hypothesis testing involves formulating the null and alternative hypotheses. In this case, the null hypothesis (Ho) posits that the average weight per bag is exactly 50 pounds, aligning with the packaging standard. Formally, Ho: μ = 50. Conversely, the alternative hypothesis (H1) argues that the true mean is less than 50 pounds, indicating a potential underfill issue, H1: μ

The selection of a significance level (α = 0.05) determines the threshold for statistical evidence needed to reject Ho. Using the t-distribution due to unknown population standard deviation and small sample size (n=20), the critical t-value for a one-tailed test at α=0.05 with 19 degrees of freedom is approximately -1.729. This critical value divides the acceptance and rejection regions for the test statistic.

Proceeding with the calculation, the sample data yields a mean weight (x̄) of approximately 49.12 pounds and a standard deviation (s) of roughly 2.15 pounds. The test statistic is calculated as t = (x̄ - μ₀) / (s / √n), which results in t ≈ -1.83. Because this value falls below the critical value of -1.729, the decision is to reject the null hypothesis.

Rejecting Ho indicates sufficient evidence to support the claim that the machinery is underfilling the packages on average. Consequently, the production process is not consistently meeting the 50-pound standard, and recalibration of the filling machinery is advisable to prevent further underfilling. Such adjustments ensure adherence to quality standards, maintain customer satisfaction, and reduce product liability risks.

In conclusion, hypothesis testing offers a systematic approach for businesses to make data-driven decisions. By rigorously analyzing sample data against established standards, companies can identify issues with production processes and implement corrective measures effectively, thereby enhancing operational efficiency and product quality.

References

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