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Unformatted text preview: Stat 340 Winter 2011 STAT 340  Winter 2011 Test 2 In this exam please assume the following:
1. All Conﬁdence Intervals and Hypothesis Tests are at a 95% and 5% level respectively.
2. You are marked according to the Clarity and completness of your solutions. 3. Functions/Distributions are strictly as deﬁned in class. 4. Failure to stop writting at the required time will result in a penalty of 5 marks. This
includes your name so write it ﬁrst. 5. The last page (leaf) of this exam is the only page you hand in.
6. Please assume that random variables are independent unless stated otherwise. 7. Watch your time. Do not waste your time on any one question. Good Luck! Extra Information Question 3: Wheel in the "wheel of fortune": "Pace 1 nf A. Stat 340 Winter 2011 Chi Squared Table 137;)! 0.001 0.005 0.01 0.025 0.05 0.1 0.5 0.9 0.95 0.975 0.99 [.995 0.993
1] 0.000 0.000 0.000 0.001 0.004 0.016 0.455 2.706 3.841 5.024 6.635 7.879 10.828
2 0.002 0.010 0.020 0.051 0.103 0.211 1.386 4.605 5.991 7.378 9.210 10.597 13.816
3 0.024 0.072 0.115 0.216 0.352 0.584 2.3 6 6.251 7.815 9.348 11.345 12.838 16.266 0
4 0.1.91 0.207 0.297 0.484 0.711 1.064 3.357 7.779 9.488 11.143 13.277 14.860 18.467
5 .210 0.412 0.554 0.831 1.145 1.610 4.351 9.236 11.070 12.833 15.086 16.750 20.515
1" 0.381 0.676 0.872 1.237 1.635 2.204 5. 48 10.645 12.592 14.449 16.812 18.548 22.458
7 0.598 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 24.322
8 0.857 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 26.124
9 1.152 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 27.877 ‘1 ._u I.“ I2” «J4
I; "3‘1 9: :«4
J:
LN 10 1.479 2.156 2.558 3.247 3.940 4.865 9.342 15.987 18.307 20.483 23.209 25.188 29.588
11 1.834 2.603 3.053 3.816 4.575 5.578 10.341 17.275 19.675 21.920 24.725 26.757 31.264
12 2.214 3.074 3.571 4.404 5.226 6.304 11.340 18.549 21.026 23.337 26.217 28.300 32.909
13 2.617 3.565 4.107 5.009 5.892 7.042 12.340 19.812 22.362 24.736 27.688 29.819 34.528
14 3.041 4.075 4.660 5.629 6.571 779013.339 21.064 23.685 26.119 29.141 31.319 36.123
15 3.483 4.601 5.229 6.262 7.261 8.547 14.339 22.307 24.996 27.488 30.578 32.801 37.697
16 3.942 5.142 5.812 6.908 7.962 9.312 15.338 23.542 26.296 28.845 32 000 34.26! 39.252
17 4.416 5.697 6.408 7.564 8.672 11.085 16.338 24.769 27.587 30.191 33.409 35.718 40.790
18 4.905 6.265 7.015 8.231 9.390 10.865 17.338 25.989 28.869 31.526 34.805 37.156 42.312
19 5.4L7 6.844 7.633 8.907 10.117 11.651 18.338 27.204 30.144 32.852 36.191 38.582 43.820
20 5.921 7.434 8.260 9.591 10.851 12.443 19.337 28.412 31.410 34.170 37.566 39.997 45.315
25 8.649 10.520 11.524 13.120 14.611 16.473 24.337 34.382 37.652 40.646 44.314 46.928 52.620
30 11.588 13.787 14.953 16.791 18.493 20.599 29.336 40.256 43.773 46.979 50.892 53.672 59.703
35 14.688 17.192 18.509 20.569 22.465 24.797 34.336 46.059 49.802 53.20? 57.342 60 275 0'1'319
40 17.916 20.707 22.164 24.433 26.509 29.051 39.335 51.805 55.758 59.342 63.591 73.402 45 21.251 24.311 25.901 28.366 30.612 33.350 44.335 57.505 61.656 65.410 69.957 73.166 80.077
50 24.674 27.991 29.707 32.357 34.764 37.689 49.335 63.167 67.505 71.420 76.154 79.490 86.661
55 28.173 31.735 33.570 36.398 38.958 42.060 54.335 68.796 73.311 77.380 82.292 85.749 93.168
60 31.738 35 534 37.485 40.482 43.188 46.459 59.3 5 74.397 79.082 83.298 88.379 91.952 99.607 85.527 90.531 95.023 100.425 104.215 112.317
96.578 101.879 106.629 112.329 116.321 124.839
' 107.565 113.145 118.136 124.116 128.299 137.208
4 118.498 124.342 129.561 135.807 140.169 149.449 70 39.036 43.275 45.442 48.758 51.739 55.329 '
80 46.520 51.172 53.540 57.153 60.391 64.278
90 54.155 59.196 61.754 65.647 69.126 73.291 8*.
100 61.918 67.328 70.065 74.222 77.929 82.358 99. u c.
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.La. Pawn '2 nf A Stat 340 Winter 2011 STAT 340 e Winter 2011 Test 2 First Name: Last Name: 1. Sheldon’s new random number generator is called ”GENARD”. He wants to know if he has created the perfect
generator. To begin with he creates 300 numbers. Each number represents the number of times (X) Sheldon
had to reprimand Penny for singing the song ” Little Kitty” incorrectly before she ﬁnally gets it right. I X 0112l3456781
[Frequency]151[823320]5350]1] Note: For this Geometric, f(av) = p (1 —p)m x : 0,1, ...and 13: (E + 1)—1 (a) [7 Marks] Using a degree of freedom of 1, stateing the hypotheses and useing a pvalue, test whether this A 1 0151+182+233+203+45+53+56+8~1 "1
data is Geometric. p 2 iii 2 —‘—_“T’—‘*— + 1
[1]
H o F is Geometric with p=0.51 Ha: F is not Geometric with p=0.51 [1]
a: Pr (X = up
0 p(1 —p)0 : 0.51 153
1 p(1 — p)1 = 0.25 75 [1]
22 1—.55—.25:0.20 72
ﬂ (6. — 0i)2
d _ Zi:1..3 ei
_ (151 — 153)2 + (82 — 75)2 + (67 — 72)2 [2]
_ 153 75 72
a: 1
p = Pr(D > d)
= (10, 50)% [1]
Therefore we have no evidence to argue the distribution is not Geometric with p:0.51. [1] (b) [2 Marks] Deﬁne what the pvalue represents in part (a). The probability we see a value as extreme as d (or worse) assuming that the null hypothesis is true. (0) [2 Marks] Does the set up you created in part (a), satisfy the requirements for the test you used in (a)? Yes  discrete data and e>=5 for all ’boxes’ "Pam: 3 m" A Stat 340 Winter 2011 2. Sheldon, next, uses his randomly generated data to create the maximum of two U(—3,2) data. The two datum
were: 0 ,—2. (a) [4 Marks] Use these datum to calculate a KS distance. x F 1?}, EL [F—FLI F—FU]
—2 gig % 0 % é — i = 046 [2] Thus, the KS distance is 0.64 [1]
0 % 1 § 5— % z 0.14 g—g (b) [2 Marks] If the distribution of the KS distances were exponential with rate 10, then determine the
pvalue in this case. Note: In reality the distribution of the KS distances is not exponential. pvalue Pr (D > d) = 1 — (1 — exp (—10  0.46)) [1]
= 0.1% [1] [I 3. Wheel of Fortune is a game show. The main attraction is a big wheel that is broken into 24 equally sized
wedges (see page 1). This wheel is spun and the corresponding money on the wedge can be won. After
100 spins, the number of times a person lands on a monetary wedge is recorded for values of $600 (16 times),
Bankrupt (5 times) and the remaining spins were some other value. (a) [4 Marks] Is the wheel landing on each space with equal probability?. Test your hypothesis.
Ho: The wheel lands on each space with equal probability. [1] Ha: The wheel does not land on each space with equal probability. 2' o p e 5 (6i — 002
d = z—
3 300 i: 61'
$600 16 a 34— 1
1 100 1632%" 2
Bankrupt 5 2—4 371— : % + [1  table]
Something Else 79 %% %g—Q : 1372 [Lang] d 2 1.372 is much smaller than we would expect from a chi squared on 2 degrees of freedom (5.99) Hence we do not reject the assumtion that the wheel lands on each space with equal probability. [1] (b) [2 Marks] How would your answer to (3) change if you used the same data but a degree of freedom of
1? So what might be said about the bankrupt wedges? REMOVED. 4. In class on Tuesday, we covered CMC integration and variance reduction. (a) [1 Mark] Why do we perform Variance Reduction?
To obtain the same accuracy with a smaller sample size. (b) [1 Bonus] In the last example on Tuesday March lst, what did we notice with regards to our answer?
Our computer couldn’t handle the sample size...it was too big. P259 4. n? A ...
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