a paper towel company's new product division creates a paper towel made from a new mix of cellulose fibers. The new paper towels cost 20% more to manufacture than the original paper towels. After testing 500 new paper towels, they find that the mean quantity of water that can be absorbed by a new paper towel is 6% more than the quantity of water that can be absorbed by an original paper towel.
A. Are the results significantly significant? Explain your answer
B. Are the results practically significant? Explain your answer
A. To determine whether the results are statistically significant, we need to perform a hypothesis test. In this case, we are comparing the mean water absorption of the new paper towels to the mean water absorption of the original paper towels.
First, we need to set up the null and alternative hypothesis:
- Null Hypothesis (H0): The mean water absorption of the new paper towels is equal to the mean water absorption of the original paper towels.
- Alternative Hypothesis (H1): The mean water absorption of the new paper towels is greater than the mean water absorption of the original paper towels.
Next, we need to select an appropriate test statistic and determine the significance level. Since we are comparing means, a two-sample t-test would be appropriate. The significance level, denoted as α, is a preset threshold typically set at 0.05 (5%).
After conducting the t-test using the provided data (mean and sample size), we can calculate the p-value associated with the test statistic. If the p-value is less than the significance level (0.05), we reject the null hypothesis in favor of the alternative hypothesis. This means that there is strong evidence to suggest that the mean water absorption of the new paper towels is significantly higher than that of the original paper towels.
B. To determine practical significance, we need to consider whether the observed difference in water absorption is meaningful or impactful in the real world. In this case, the test results show that the new paper towels can absorb 6% more water than the original ones on average.
To assess practical significance, we can consider factors such as consumer preferences, cost considerations, and market demand. If the 6% increase in water absorption has a substantial impact on the product's performance or offers a significant advantage over competitors, then the results could be considered practically significant.
However, if the cost of manufacturing the new paper towels is significantly higher (20% increase) and the observed increase in water absorption does not provide sufficient practical benefits or advantages, the results may not be practically significant. It ultimately depends on the specific context and goals of the paper towel company's new product division.