1. This is the first question in a three-part sen'es where we will be studying the same population. 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 confidence intervals we construct. After a particularly bad exp-en'ence with a large coffee that seemed to be overdosed with caffeine, we gaggle to investigate the caffeine content of large coffees sold by our favorite coffee franchise. A colTee experience engineer from the companyr decides to participate in the study, and they reveal that the true {population} standard deviation of caffeine content in a large coffee is o = 25 mg. (T hey 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=5o, if =1lo mg.s=27 mg Here are some facts about the standard normal distribution that could prove useful: Pr: 5 1.945} = .95, Pt: s1.9eoi=.975.Ptz s 2.53} = .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 unspecified u.) b] Making use of our sample data as well as the population standard deviation value 0, construct a 95% confidence 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 o is known.) c} Which of the provided sample statistics was least useful for parts alt-c)? Explain why this information wasn't helpful. d] Did your condence interval include 100', the value the engineer claimed was correct for the population mean? (Note that we will use an argument based on this observation to motivate hypothesis testing later on) 4. ISuppose we wish to know the the population proportion of students at the University of Watertoo who respond yes to the following question: \"Have you. at any point in your life. amassed a collection of digital monsters (of any franchiser'species) while playing a video game?\" We collect a suitably randomized sample of size n = 30 and record a total of 18 "yes" responses. Here are some facts about the standard normal distribution that could prove USSfUII PtZ 51.645}=.95.P(Z S 1.960) = .975r PtZ S 2.58} = .995 a) Construct a 90% condence interval for the true population proportion, p: without using the \"plus four\" correction b] Construct a 9MB condence interval for the true population proportion, p, but this time, use the \"plus four" conecon. c} The president of the university has taken an interest in our study, Seeking perfect information, the president requests that all students at the University answer the digital monsters question. As a result of this megaproject, the true population proportion is revealed to be p = .751 a value outside both of the condence intervals we've constructed so far. Assuming our sampling strategy was appropriate, how can we reconcile this information about the ane parameter value with the information from our sample? Using what you know about condence intervals. explain how it is possible for the the population parameter to fall outside of our calculated intervalis). 2. Let's return to the same caffeine content example from question 1. but this time we'll behave as 'rf the population standard deviation 0. is unknown. 1ll'tle'll use the same sample as before: a sample of 50 large coffees produces the following sample statistics: n=50_ $2110mg,s=2?mg Here are some facts about the Student's t-distn'bution with 49 degrees of freedom that could prove useful: Pft s 1.63) = .95, P(t s 2.01) = .915, Pa 5 2.68] = .995 a) Using the information in this sample, construct a 95% condence interval for the true population mean. u. Note that we do not know 0 any more. so we must use a tdistribution to construct this interval- b) Is the width of this interval larger or smaller than the condence interval from Question 1? Why might these intervals be different sizes? c] Did yOur condence interval include 'il'1 the value the engineer claimed was correct for the population mean? (Again, note that we will use an argument based on this observation to motivate hypothesis testing later in the course.) 3. For the third part in this series. we're still going to investigate the cal'leine content for these coees, but this time we're going to be cheap and only collect a sample of size n = 5. However, let's assume that the other sample statistics happened to turn out the same for this sample. Like question 2, we'll once again assume 0 is not known. our new sample statistics are; n=5, $=110mg,s=27mg Here are some facts about the Student's tdistribution with 4 degrees of freedom that could prove useful: PH 5 2.132) = .95. P'{t S 2.???) = .9?5. P(t'S 4.60) = .995 a] Using the information in this sample. construct a 95% condence interval for the true population mean, p. b] Is the width of this interval larger or smaller than the condence interval from Question 2? Why might these intervals be different sizes? c} Did your condence interval include 100, the value the engineer claimed was correct for the population mean? (Once again, note that we will use an argument based on this observation to motivate hypothesis testing later on)