The test
StatsTest2.DVI Intro to Statistics Test 2 April 16, 2020 L. Lehman You may use a calculator, the table of the normal distribution, the t-distribution table, and a sheet of your own notes, prepared before you begin the test. Use of the textbook, the lecture notes, or assignment solutions posted on Canvas is not allowed. There is no time limit (other than the time the test is due), but once you begin, do not consult any outside sources (in person or online). The test is due through Canvas or as a email attachment to me (
[email protected]) by 11:59 pm today (Thursday). Before turning your test in, print your name and sign below to pledge that you have not given or received unauthorized aid on the test. Print Name: Signature: Questions 1–5 refer to the following experiment. A vase contains twelve marbles—five are red and seven are black. Three marbles are drawn from the vase in succession, and the color of each marble is recorded. (The marbles are not returned to the vase before the next marble is drawn.) Answer the following questions, assuming that on each draw, all the marbles currently in the vase have equal probability of being selected. (Write each probability as a fraction, rather than in decimal form. You do not need to reduce the fractions to lowest terms. Hint: A tree diagram might be helpful here. Explaining your answers that way or otherwise might result in partial credit.) (6) 1. What is the (conditional) probability that the second marble chosen is red, given that the first marble chosen is red? (6) 2. What is the (conditional) probability that the third marble chosen is red, given that the first and second marbles chosen are red? (6) 3. What is the probability that the all three marbles chosen are red? (6) 4. What is the probability that none of the three marbles chosen is red? (6) 5. What is the probability that exactly one of the three marbles chosen is red? In Questions 6–8, suppose it is known that the number of high cards (aces, kings, queens, or jacks) in a five-card (poker) hand has a distribution that is skewed to the right, with mean µ = 1.55 and standard deviation σ = 1.00. For a random sample of n = 100 poker hands, each dealt from a well-shuffled 52 card deck, let x be the mean of the number of high cards in those hands. Answer the following questions about the distribution of sample means x in all possible samples of 100 poker hands. (You may assume that this sample size is large enough so that the Central Limit Theorem applies.) (5) 6. What is µx, the mean of the distribution of these sample means? (5) 7. What is σx, the standard deviation of the distribution of these sample means? (4) 8. Which of the following is the best description of the shape of the distribution of these sample means? (a) It is skewed to the right (the same as the population). (b) It is skewed to the left (the opposite of the population). (c) It is approximately normal. (24) 9. The numbers x in the following table represent the net weight of six bags of Doritos. x x − x (x − x)2 29.4 28.8 28.9 29.1 29.3 29.7 a) Calculate the mean x and standard deviation s of this data set. Show your work by filling in the table above, and write your answers in the blank spaces below. x = s = b) Fill in the blanks below to find a 95% confidence interval for µ, the mean of the population of net weights for all similar bags of Doritos. (Make the boundaries of the interval accurate to two decimal places. You may assume that the population is approximately normal and that the six bags represent a random sample chosen from that population.) < µ="">< in questions 10–14, suppose that an over-the-counter pain medicine has proven to be effective in 70% of users. researchers at the company believe that they have found a more effective formula, and plan to run a hypothesis test, with a significance level of α = .01, on a random sample of n = 200 pain medicine users. let p̂ be the proportion of users in the sample who find the new formula effective, and let p be the proportion in the population of all potential users who would find the new formula effective. answer the following questions. (8) 10. what are the null and alternative hypotheses for this test? (write your answers in terms of the appropriate value, p or p̂.) (8) 11. assuming that the null hypothesis is correct, what are the mean, µp̂, and standard deviation, σp̂, of the data set of p̂ values over all possible samples of 200 pain medicine users? (calculate σp̂ to three decimal places.) (8) 12. if 152 of the 200 people in the sample report that the new formula is effective, should the null hypothesis be rejected or not (with significance level of α = .01)? show your work and explain your answer. (8) 13. if the null hypothesis is correct, what is the probability that, for a sample of n = 200 people, p̂ is larger than 0.76? (make your answer accurate to two decimal places. hint: use the normal distribution table and the data from question 11.) 14. extra credit question (8 points maximum): suppose that in fact p = .8 (that is, the new formula is actually effective in 80% of pain medicine users). under that assumption, what is the probability that, for a sample of n = 200 people, p̂ is smaller that 0.76? (to receive full credit, show your work and explain how you find your answer. again, make your answer accurate to two decimal places.) in="" questions="" 10–14,="" suppose="" that="" an="" over-the-counter="" pain="" medicine="" has="" proven="" to="" be="" effective="" in="" 70%="" of="" users.="" researchers="" at="" the="" company="" believe="" that="" they="" have="" found="" a="" more="" effective="" formula,="" and="" plan="" to="" run="" a="" hypothesis="" test,="" with="" a="" significance="" level="" of="" α=".01," on="" a="" random="" sample="" of="" n="200" pain="" medicine="" users.="" let="" p̂="" be="" the="" proportion="" of="" users="" in="" the="" sample="" who="" find="" the="" new="" formula="" effective,="" and="" let="" p="" be="" the="" proportion="" in="" the="" population="" of="" all="" potential="" users="" who="" would="" find="" the="" new="" formula="" effective.="" answer="" the="" following="" questions.="" (8)="" 10.="" what="" are="" the="" null="" and="" alternative="" hypotheses="" for="" this="" test?="" (write="" your="" answers="" in="" terms="" of="" the="" appropriate="" value,="" p="" or="" p̂.)="" (8)="" 11.="" assuming="" that="" the="" null="" hypothesis="" is="" correct,="" what="" are="" the="" mean,="" µp̂,="" and="" standard="" deviation,="" σp̂,="" of="" the="" data="" set="" of="" p̂="" values="" over="" all="" possible="" samples="" of="" 200="" pain="" medicine="" users?="" (calculate="" σp̂="" to="" three="" decimal="" places.)="" (8)="" 12.="" if="" 152="" of="" the="" 200="" people="" in="" the="" sample="" report="" that="" the="" new="" formula="" is="" effective,="" should="" the="" null="" hypothesis="" be="" rejected="" or="" not="" (with="" significance="" level="" of="" α=".01)?" show="" your="" work="" and="" explain="" your="" answer.="" (8)="" 13.="" if="" the="" null="" hypothesis="" is="" correct,="" what="" is="" the="" probability="" that,="" for="" a="" sample="" of="" n="200" people,="" p̂="" is="" larger="" than="" 0.76?="" (make="" your="" answer="" accurate="" to="" two="" decimal="" places.="" hint:="" use="" the="" normal="" distribution="" table="" and="" the="" data="" from="" question="" 11.)="" 14.="" extra="" credit="" question="" (8="" points="" maximum):="" suppose="" that="" in="" fact="" p=".8" (that="" is,="" the="" new="" formula="" is="" actually="" effective="" in="" 80%="" of="" pain="" medicine="" users).="" under="" that="" assumption,="" what="" is="" the="" probability="" that,="" for="" a="" sample="" of="" n="200" people,="" p̂="" is="" smaller="" that="" 0.76?="" (to="" receive="" full="" credit,="" show="" your="" work="" and="" explain="" how="" you="" find="" your="" answer.="" again,="" make="" your="" answer="" accurate="" to="" two="" decimal="">