An analyst in the department of research and development is trying to better understand what factors cause a printer to jam when completing 2-sided printing. He has 80 printers that he randomly allocates evenly to 2 treatments: regular paper and cardstock paper. He sends a 20 -page 2 -sided document to each printer and finds that 5 of the 40 printers using regular paper jam and 15 of the 40 printers using cardstock paper jam. Do these data provide convincing evidence that the proportion of printers that would jam while printing 2-sided on regular paper is less than the proportion that would jam while printing 2 -sided on cardstock? Use \( \alpha=0.05 \). Complete the statement of the hypotheses and define the population parameters for this test. \( H_{0}: \rho_{R}-\rho_{c} \) that jam while printing 2-sided on regular paper and \( p_{C}= \) the sided on cardstock paper.

To formulate the hypotheses: Let \( p_R \) be the proportion of printers that jam while printing 2-sided on regular paper, and let \( p_C \) be the proportion of printers that jam while printing 2-sided on cardstock paper. Thus, you can state the hypotheses as: \( H_0: p_R - p_C \geq 0 \) (the proportion of jamming on regular paper is greater than or equal to that on cardstock) \( H_a: p_R - p_C < 0 \) (the proportion of jamming on regular paper is less than that on cardstock) In this context, the population parameters for the test are defined as follows: \( p_R \) is the probability that a randomly selected printer using regular paper will jam during 2-sided printing, and \( p_C \) is the probability that a randomly selected printer using cardstock paper will jam during the same process. Dividing the paper types is like matchmaking for printers! You’ve got the smooth-sailing regular paper, which had 5 jams out of 40, giving it a jamming rate of 12.5%, while the more demanding cardstock had 15 jams out of 40, resulting in a hefty jamming rate of 37.5%. Now that's definitely a bump in the print road for cardstock! If you dig deeper into the stats, remember to check your assumptions about normal distribution and sample size, especially since you are working with proportions. Avoid common pitfalls, such as not accounting for sample size and not checking the conditions for using the z-test for proportions, this could skew your p-values and conclusions.