Test 2 for MS 5312
Time allowed: one hour
Date: 14 November 2008
Total: 50 marks
Attempt all questions.
Correct your answers to at most 4 decimals, when applicable.
Question 1 (18 marks)
(a) Each day, the United States Customs Service has historically intercepted about $28 million in contraband goods being smuggled into the country, with a standard deviation of $16 million per day. On 64 randomly chosen days in 2007, the US Customs Service intercepted an average of $30.3 million in contraband goods.
(i) Does this sample indicate (at a 5% level of significance) that the Customs Commissioner should be concerned that smuggling has increased above its historic level? (6 marks)
(ii) What is the p-value? Do you make the same decision as in subpart (i) if you use the p-value approach? (3 marks)
(iii) Construct a 95% confidence interval for the population mean. (5 marks)
(b) Define the probability of type I error and that of type II error . (4 marks)
Question 2 (20 marks)
Obama has won the American presidential election. Suppose that you conducted a public opinion poll one month before the outcome of the election, and randomly selected 100 American citizens to be surveyed their choices between Mccain and Obama. The survey showed that 65 citizens would choose Obama and 35 Mccain.
(a) Assume that the population proportion of choosing Obama was 60%. Determine the mean and standard error of the sampling distribution of the proportion. (4 marks)
(b) Construct a 90% confidence interval for the population proportion of choosing Obama, based on the survey result. (5 marks)
(c) Did you have sufficient evidence, based on the survey result, to reject the assumption made in part (a) at a 5% level of significance? (6 marks)
(d) Suppose that you required 95% confidence in estimating the population proportion of choosing Obama with error within 0.03. You guessed that this proportion would be between 40% and 75% inclusive. What is the sample size? (5 marks)
Question 3 (12 marks)
One hundred athletes are selected to conduct a drug test just after their matches. Suppose that 80 athletes did not take any drugs while 20 indeed took some drugs. Among the 80 nondrug-taking athletes, only one athlete yields a false positive test result (so-called a false positive). Among the 20 drug-taking athletes, 18 show true positive test results (so-called true positives).
(a) What is the probability of a positive (which is either false or true)? (3 marks)
(b) If one athlete is randomly chosen from these 100 athletes, what is the probability of finding a drug taker and a positive? (2 marks)
(c) If an athlete shows a positive, what is the probability of a drug taker? (2 marks)
(d) Consider only the first 10 athletes, what is the probability of observing at least two positives among them? (5 marks)
===
Solution to Question 1:
(a,i) H0 : = 28
H1 : > 28 [2]
[2]
z0.05 = 1.645 which is greater than 1.15 [1]
We do not reject H0 at = 5% [1]
(a,ii) p-value = 0.1251 [2]
which is greater than 0.05.
We make the same decision, i.e. We do not reject H0 at = 5% [1]
(a,iii) The 95% confidence interval for μ is
[3]
or [2]
(b) = Pr(rejecting H0 H0 is true) [2]
= Pr(not rejecting H0 H0 is false) [2]
Solution to Question 2:
(a) [2]
[2]
(b) [1]
The 90% confidence interval for p is
[3]
[1]
(c) H0:
H1: [1]
The observed value of the test statistic is
[2]
From the z-table, [1]
Since 1.0206 < 1.96, we do not reject H0 at = 0.05. [1]
Thus, there is no sufficient evidence to reject the assumption in part (a). [1]
(d) Because , to determine the sample size in a conservative way, we select 0.5 [2] to maximize in the sample size calculation formula.
[3]
Solution to Question 3:
(a) Let B be the event "a positive". Then the total probability law yields
[3]
(b) [2]
(c) Let A be the event "a drug taker". Then Bayes' law yields
[2]
(d) Let x be the number of positives. Then the binomial probability formula gives
[1]
[2]
[2]
五、北京權者 重重設障
但正如北京當局在六四慘案後重重設障，03年到今天，中央政府也繼續給議會的民主派出難題，繼續防範香港的民主發展對一黨專政造成衝擊。
它採用的策略，除了在經濟上給港人甜頭外，也在某幾項實質的民憤事件上順應香港的民情，這也可被視為正常的拆彈動作，將一切具爆炸性、可令港人上街的事件清除。除了讓葉劉淑儀倉皇出走，老董「腳痛」下台外，也讓西九推倒重來，讓紅灣半島不拆；讓楊永強辭職……
然而人大卻突然於04年釋法，封殺0708年 雙普選，繼而在07年，再用「決定」去扼殺2012年雙普選。整體局面可說是一種懷柔式的封殺 (killing you softly, with his bait) 。明顯這是中央政府從背後發功以連消帶打、釜底「加」薪的方式回應香港市民的訴求，將訴求的實物目標放給港人，但卻保留，甚至進一步加強訴求底下《基本法》的政治枷鎖。
本來，《基本法》附件二規定，2007年 後香港特區立法會的產生辦法，法案、議案的表決程序，如需對附件的規定進行修改，只須經立法會全體議員三分之二多數通過，行政長官同意，並報全國人民代表大會常務委員會備案就可以。
上述《基本法》條文賦予港人爭取0708年 雙普選的空間，於是爭取07年普選行政長官和08年普選立法會，成為了民主運動的主軸，甚至保皇黨，如民建聯及自由黨亦曾將0708年雙普選寫入其黨綱中。
但中央政府公然違背「港人治港、高度自治」的承諾，於2004年4月6日，出其不意地以釋法之名義改動《基本法》。
人大釋法將修改《基本法》的權力完全收歸人大常委，並於4月26日通過決議，剝奪《基本法》訂明港人在0708年應得的 民主普選權利，還讓特區政府其後推出一個既無時間表復缺路線圖的05年政制「發展」方案，以各種公開的壓力及暗中的利誘，企圖令民主派接受，造成港人甘願屈服的既成事實。若非港人在關鍵時刻站出來令民主派歸隊，這個如雞肋般的05年政制「發展」方案，早已填鴨般填進港人的喉頭之中！
但2005年 的政改方案遭否決，特區及中央並一眾親共輿論一直將政制發展「原地踏步」的責任推給民主派，這是一種「明屈」的伎倆，類近於新疆當局「招呼」本港記者的做法。
民主派卻苦無良策予以有力的駁斥。面對沒有0708年 雙普選的既成事實，民主派只好退而求其次，祭出爭取2012年雙普選的旗號，以向市民交代。但旗幟雖在，爭取的策略和勢頭卻付之闕如。
中央政府食髓知味，眼見泛民主派對扭曲式釋法沒有作出有力的反抗，於是又再於2007年12月30日，人大常委再作出「決定」，否決2012年 雙普選，但為了緩和港人一而再被侮辱與愚弄所產生的可能反彈，中央政府虛與委蛇，拋出了所謂容許港人於2017年 及2020年進行雙普選的「期票」，
但同時又再一次要求民主派合作，將2017年 及2020年雙「普」選能否兌現，與2012的政制「發展」方案能否被民主派接受掛鈎，這無異又是再一次將民主派推到台前，以一個空泛的2017年及2020年承諾，迫使其盡早投降。
香港早就具備雙普選的條件，擁有普及的教育制度、健全的司法制度、自由開放的公共空間等，早於1988年便應實行普選，但是港人的民主訴求一再被打壓，1988年一拖20年至2007年及2008年，現在中央又再拖延十年有多至2017年及2020年，香港人還有幾多個十年？
事實擺在眼前，若無法在爭取2012雙普選方面有突破口，港人只會被陰乾同化成一群順民；若無法找到突破口，香港的民主派只會遲早被中和掉 (neutralized) ，雙普選只會不斷被拖延戰術玩弄，香港永無普選的日子。這是中央政府自03年到今天，給民主派出的大難題。
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