[7] [2] 1. This is the rst question in a threepart series where we will be studying the same pop ulation. However, as we proceed from question 1 through to question 3, we will afford ourselves progressively less information. Let's explore the impact this has on the resulting condence intervals we construct. After a particularly bad experience with a large coffee that seemed to be overdosed with caffeine, we decide to investigate the caffeine content of large coffees sold by our favorite coffee franchise. A coffee experience engineer from the company decides to participate in the study, and they reveal that the true (population) standard deviation of caffeine content in a large coffee is a = 25 mg. (They also claim the population mean is ,u = 100 mg, but we're not quite ready to believe it.) We collect a sample of 50 large coffees and record the following sample statistics: n=50, i=110mg, s=27mg Here are some facts about the standard normal distribution that could prove useful: P(Z g 1.645) = .95, P(Z g 1.960) = .975, P(Z g 2.58) = .995 a) Let X be the random variable whose outcomes represent possible sample means. State the probability distribution for X, the sampling distribution of the mean, and the parameter values we would expect according to the Central Limit Theorem. (For the population mean, leave the value as an unspecied u.) [2] b) Making use of our sample data as well as the population standard deviation value 0, construct a 95% condence interval for the true population mean. (Note that we can use a normal distribution to determine the error bound of the mean because we're in the unusual situation where or is known.) c) Which of the provided sample statistics was least useful for parts a)-c)? Explain why this information wasn't helpful. d) Did your condence interval include 100, the value the engineer claimed was cor- rect for the population mean? (Note that we will use an argument based on this observation to motivate hypothesis testing later in the course.)