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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 Introductory Statistics December 29, 2005 Time: 2:30 p.m.  4:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal
culator which fulﬁls the following criteria: (a) it should be selfcontained, silent,
batteryoperated and pocketsized and (b) it should have numeraldisplay facilities
only and should be used only for the purposes of calculation. It is the candidate’s responsibility to ensure that the calculator operates satisfactorily
and the candidate must record the name and type of the calculator on the front page of the examination scripts. Lists of permitted/prohibited calculators will not be made
available to candidates for reference, and the onus will be on the candidate to ensure
that the calculator used will not be in violation of the criteria listed above. Answer ANY FOUR questions. The questions are of equal value. 1. The following data show the number of hours (X) of 10 randomly selected
students who studied for a statistics test and their scores (Y) on the test: In
IE! (a) Fit a simple linear regression model Y = ,60+;81$+€ to the data. (b) Given that SSE _ zyzlm — 2)? = 234.3128 and 3:10;. — Xi)? =
376.0. 1. Estimate with 0.95 conﬁdence the score that an individual student
will obtain in the test if he studies 28 hours. Is this estimate reliable?
Why or why not? ii. Interpret 30. Do you think it is reasonable to test the hypotheses that Ho : [30 = 0.0 against H1 : ﬁg > 0.0? Do not attempt to carry
out the test, but just explain brieﬂy. iii. Test H0 : 51 = 3.0 against H1 : [31 > 3.0 at a = 0.05. State the name
of the test, the test statistic, the rejection region and your conclusion
clearly. S&AS: STAT1306 Introductory Statistics 2 2. A Statistics course has 2 subclasses, namely A and B, that use identical hand
outs, assignments, class tests and examinations, but are taught by professors
Audrey and Betty, respectively. Audrey has been the teacher of this course
for years, and Betty is a new teacher. The Head of Department and the Chair
Professor suspect that teacher Betty may not be a good teacher and hence her
students may suffer and their ﬁnal results may not be as good as the students in class A. (a) To justify his suspicion, the Head of Department draws a random sample
10 students from each of the two classes with their ﬁnal marks recorded below. Carry out an appropriate test for the Department Head’s claim at the
0.05 level of signiﬁcance. State the appropriate null and alternative hy
potheses, the name of the test, the test statistic, the rejection region and
your conclusion clearly. (b) The Chair Professor studies this problem from another perspective by
comparing the failure rates of the two subclasses. He randomly selects
200 students from the 2 classes and observes 12 failures among the 120 students from class A and 10 failures among the 80 students from class
B. 1. Construct a 90% conﬁdence interval for the overall failure rate of the
course by combining the two subclasses. ii. Test the hypothesis that the failure rate of class B is higher than that
of class A at the 0.05 level of signiﬁcance. State the appropriate null
and alternative hypotheses, the name of the test, the test statistic,
the rejection region and your conclusion clearly. iii. Are your conclusions in (a) and (b)(ii) consistent? Which test do you
think is more appropriate to detect a difference in the teaching skills
between a new teacher and an experienced teacher? Brieﬂy describe
in not more than 100 words. S&AS: STAT1306 Introductory Statistics 3 3. From long experience with a process of manufacturing gunpowder, it is known
that the resulting muzzle velocity, Y, is normally distributed with a mean 1000
m/ sec and a standard deviation of 100 In / sec. A proposal for modiﬁcation is
received for which it is claimed that the new gunpowder will result in a faster
muzzle velocity, but leaving the standard deviation unchanged. (a) Formulate the null and alternative hypotheses for the above claim. (b) The new gunpowder is to be tested in n = 15 shells with mean muzzle
velocity 17. Let a = P(Committing a type I error) which is now ﬁxed at
the 0.025 level. Construct a rejection region so that the null hypothesis
is rejected if 17 falls in this region. (c) If the true average muzzle velocity for the new gunpowder is ,u = 1080
m/sec, with the conditions in (b) remained unchanged, ﬁnd [3 where
)6 = P(Committing a type II error). (d) What is the smallest value of n required in order to make 0: = 0.025 and
ﬂ S 0.025 for the above hypothesis testing procedure? 4. The probability distribution of X, the daily number of passengers on a heli~
copter shuttle run from Hong Kong to Macau, is given as follows: I“
I Assume that X is independent from day to day. (a) What is the probability that the ﬁrst day having 3 or more passengers
will occur on the fourth day starting from today? (b) What is the probability that in a particular week (7 days), the helicopter
shuttle will have three or more passengers in fewer than two days? (c) What is the approximate probability that in a particular year (365 days),
the helicopter shuttle will have at most one passenger in fewer than 150
days? ((1) What is the probability that the total number of passengers in two con
secutive days is equal to 3? (e) It is known that the total number of passengers in two consecutive days
equals 3. What is the probability that there is exactly one of these days
with no passenger? (f) Find the mean and the variance of the daily number of passengers on the
helicopter shuttle. (g) In a random sample of n = 100 days, the daily number of passengers
is recorded. What is the approximate probability that the sample mean
exceeds 2.0? S&AS: STAT1306 Introductory Statistics 4 5. (a) Two scales, A and B, are used in a laboratory to weigh the rock specimens
(in grams). A random sample of 10 rock specimens was selected and the
weight of each rock specimen was obtained from each of the two scales.
The following data were obtained in an experiment to check whether
there is a systematic difference in the weights (in grams) obtained from
the two scales A and B: Rock Specimen Scale A (X) VOOONCDCﬂibOOMl—‘I 10 Deﬁne Di = X, — Yi, the difference of the weights obtained from scales A
and B. i. Test HOZUD=0.25 VS H110D<0.25 at the 0.05 level of signiﬁcance where an is the population standard ‘ deviation of the difference Di. Write down clearly the test statistic,
the rejection rule and your conclusion. Also state the assumption(s)
necessary for the test to be valid. ii. With the conclusion and assumption(s) in (i), construct a 95% con
ﬁdence interval for up, the mean of the differences of the weights
obtained from the two scales. Is there evidence to indicate that the
weights obtained with the two scales are signiﬁcantly different? (b) Let X be a continuous random variable with probability density function 0
(1+¢)2, where C is an appropriate constant. i. Find C. ii. Deﬁne a new variable Y = X / (1 + X Show that the distribution
function of Y is F(y) = y for 0 < y < l and hence ﬁnd P(Y < 2/5). 0<x<oo;
f($) = elsewhere. S&AS: STAT1306 Introductory Statistics 5 A LIST OF STATISTICAL FORMULAE 2. Chebyshev’s Theorem: P( X — u < k0) _>_ 1 — £2— for k > 1 3. P(A u B) = P(A) + P(B)  P(A n B)
4. P(A n B) = P(B)  P(AIB) = P(A) . P(BA) 5. Bayes Theorem I P(AJ'IB) = i=1 2 . i 7 Va?"(Y) = E[(Y  W] = 2(31— u)2p(y) = E(Y2)  #2 = 02 9' X N :10 = _p)n—ZI forx=0)1)"'7n; = np,Var(X) = np(1 .._p); MX(t) 2 (p ct +q)n
10. X N Geometric(p), P(X = x) = p (1 _ p)z—1’ for x = 1, 2, _ _ .; E(X) = 1/p,var<x> = (1 P)/P2;Mx(t) = 13:6) 11 XNNeQBin(T,P), P(X=95) =( )Pr (1—p)$_', forx=r,r+1,r+2,~ r—l M) = r/p, Vamx) = «1 —p)/p2;MX(t) = [ p at 12. X N P0isson(A), P(X = x) = %e"‘ for x = 0,1,  ; E(X) = Var(X) = A Y— f —
13. Y ~ N(u,a2), z = w = Y " ~ N(0,1)
Var(Y) 0 14. E(X) = p, Var(X) =0;
X—u
0/\/71 15. Central Limit Theorem : For n being large, Z = xv mo, 1) _XM 16.Z a/ﬁ ~ N(0,1); Xiza/za/Jﬁ S&AS: STAT1306 Introductory Statistics 6 X—u _
 = N n—; Xitn— a
17 T S/ﬂ t 1 1,/2S/\/ﬁ
151) . A 15(125)
18. Z=———————~N(0,1); pizaz
\F—pu—pvn / n
X_' _ _
19_ Z=L__}_/)_2_ﬂ£2___mil~]v(0,1)
3+1)”.
’nx ny (nx — 1) + (ﬂy  1) (nx + ny — 2)
X—Y’ —— — — — 1 1
21 T = (——S—)—M ~ tnx+ny—2; (X “ Y) itnx+ny—2,a/2 SP —— + ——
P L L nx 7;,»
"X "Y 20. Pooled Variance: $12, = + A 22_ Z = W ,;, N(0,1); 151(1 ' 151) + 152(1 132) A _ A i z _
[111(1—1712 + 7120—112) (p1 p2) a/z n1 n2
n1 n2 2 __ (n — 1)S2 2 . (n — 1)S'2 (n —1)S2 X — 02 X(df=n—1)7 ﬂ '
24. = n nz2i=1— (21:1 12:1 n
V n 21:1 Xi — ( i=1 Xi)2V ” 22:1 Y? ‘ (21:1 YDZ
[Bx/n — 2
25. T = ——; ~ td.f..—_n—2
\/1  p2
A 2711(Xi—XXYE—Y) A — A 
26. = L————_——' = ——
‘31 E?=1(Xi J02 ’ ° Y 51"
m a SSE
27. SSE: Yi—Yi2; 2:
. 1 X2 A 2
28.ﬁ~N(ﬂ,a§(—+—n———)); ~N( __"e__.._.
0 0 n 21:1(371 "’ X)2 ’81 ’81, 22:1(372' —‘ XV 3:2) I}: tn_2,a/2 X = [130); ?o It tn_2,a/2 X 3.6.(Y2'0lX = $0). ...
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 Normal Distribution, Null hypothesis, Probability theory, probability density function, Type I and type II errors

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