Write your answers onto one or more separate sheets of paper. Clearly mark each sheet with your name and your lab time. Show your work and round all answers to at least 3 decimal places.
1. The population distribution of weights of “princess beans,” Phaseotus vulgaris, is known to be normally distributed with 𝜇=485 mg and 𝜎=112 mg. Suppose you randomly select 3 beans from this population.
a. What is the probability that all 3 beans weigh between 400 and 600 mg? (4pt)
b. What is the probability that the mean weight of the 3 beans is between 400 and 600 mg? (4pt)
c. Suppose that you knew that the population distribution of weights was in fact positively skewed, but still had the same mean and standard deviation. Would you expect this to change your results in part (b)? Why/why not? (2pt)
d. Of the probabilities you calculated in parts (a) and (b), which is larger? Regardless of the population distribution parameters, explain why this must will always be the case. (2pt)
2. A dairy researcher has developed a new technique for culturing cheese that supposedly ages the cheese faster than the traditional methods, without affecting other properties like texture or taste. To implement this new technique, the cheese company would need to initially spend millions of dollars retrofitting their manufacturing plant. But if it does actually reduce the aging time of their cheese, even marginally, it will lead to higher profits for the company over time.
In an experiment the researcher uses both methods of culturing cheese to see if there is a statistically significant reduction in aging time with the new method.
a. Write an appropriate null and alternative hypothesis for this experiment. (2pt)
b. In the cheese company’s perspective, explain what the consequence of a Type-I error would be. (2pt)
c. In the cheese company’s perspective, explain what the consequence of a Type-II error would be. (2pt)
d. Suppose the researcher analyzed the data and found a p-value of 0.18. What would be the appropriate conclusion, in the context of the data? (2pt)