A Laundry Detergent Company Wants To Determine If A New Formula Of Detergent, A, Cleans Better Than The Original Formula, B. Researchers Randomly Assign 500 Pieces Of Similarly Soiled Clothes To The Two Detergents, With 250 Pieces In Each Group. After

**A Laundry Detergent Company's Experiment: A Statistical Analysis**

A laundry detergent company wants to determine if a new formula of detergent, A, cleans better than the original formula, B. Researchers randomly assign 500 pieces of similarly soiled clothes to the two detergents, with 250 pieces in each group. After washing and drying, the researchers measure the cleanliness of each piece of clothing using a standardized scale. The company wants to know if the new formula, A, is significantly better than the original formula, B.

The company's researchers are faced with a classic problem in statistics: comparing the means of two groups. They want to determine if the new formula, A, has a significantly higher mean cleanliness score than the original formula, B. To do this, they need to perform a statistical analysis of the data.

Q: What type of statistical test should the researchers use to compare the means of the two groups? A: The researchers should use a two-sample t-test to compare the means of the two groups. This test is suitable for comparing the means of two independent groups, and it assumes that the data follows a normal distribution.

Q: What are the assumptions of the two-sample t-test? A: The two-sample t-test assumes that the data follows a normal distribution, and that the two groups are independent and randomly sampled. It also assumes that the variances of the two groups are equal.

Q: How do the researchers determine if the new formula, A, is significantly better than the original formula, B? A: The researchers will calculate the p-value of the two-sample t-test, which represents the probability of observing the difference in means between the two groups, assuming that there is no real difference. If the p-value is less than a certain significance level (e.g. 0.05), the researchers can conclude that the new formula, A, is significantly better than the original formula, B.

Q: What is the significance level, and why is it important? A: The significance level is a threshold for determining whether the results of the statistical test are statistically significant. In this case, the researchers will use a significance level of 0.05, which means that if the p-value is less than 0.05, they can conclude that the new formula, A, is significantly better than the original formula, B. The significance level is important because it helps to prevent false positives, which occur when a statistically significant result is observed by chance.

Q: What are the potential limitations of the two-sample t-test? A: The two-sample t-test assumes that the data follows a normal distribution, which may not always be the case. Additionally, the test assumes that the variances of the two groups are equal, which may not always be true. If these assumptions are violated, the results of the test may not be reliable.

Q: What are some alternative statistical tests that the researchers could use? A: If the assumptions of the two-sample t-test are violated, the researchers could use alternative statistical tests, such as the Wilcoxon rank-sum test or the permutation test. These tests are non-parametric, meaning that they do not assume a normal distribution, and they can be used to compare the means of two groups.

In conclusion, the researchers can use a two-sample t-test to compare the means of the two groups and determine if the new formula, A, is significantly better than the original formula, B. However, they need to be aware of the assumptions of the test and potential limitations, and consider alternative statistical tests if the assumptions are violated.

• Hogg, R. V., & Tanis, E. A. (2001). Probability and Statistical Inference. Prentice Hall.
• Kutner, M. H., Nachtsheim, C. J., & Neter, J. (2004). Applied Linear Statistical Models. McGraw-Hill.
• Wilcox, R. R. (2005). Introduction to Robust Estimation and Hypothesis Testing. Academic Press.