Download SOLUTIONS!!!! STATS TEAM TBT Feb 2013

TBT Invitational 2013
Question #1
Statistics Team
Let X be the first ten whole numbers (in order) and let Y be the first ten triangular numbers
A=the correlation coefficient of Y on X.
X
Y
s
C= Y
sX
B=
D=the x-intercept of the regression line between X and Y.
TBT Invitational 2013
Question #2
Statistics Team
Let X be normally distributed with mean 5 and variance 4, and Y be normally distributed with
mean 2 and standard deviation 2. Assume X and Y are independent variables. Round your answers
to four decimal places.
A= P  X  7.5
B= P Y  0 | Y  1
C= P  X  6 | Y  6 
D= P Y  X 
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Question #3
Statistics Team
Topic: Psychology- A clinical trial was performed with antidepressant drugs as treatment and
researchers observed whether or not the patient relapsed. The table below shows the proportion
of subjects with their drug and response:
Response/Treatment
Imipramine
Lithium
Imipramine +
Placebo
Lithium
Relapse
0.120
0.087
0.146
0.160
No Relapse
0.147
0.166
0.107
0.067
A=Probability that a subject used lithium (alone or in combo) and did not relapse
B=Probability that a subject relapsed on Imipramine.
C=Probability that a subject relapsed, given that they used only Imipramine.
D=Probability that a patient was on a placebo, given that they relapsed.
TBT Invitational 2013
Question #4
Statistics Team
Topic: Social sciences- 9% of Americans cannot find the United States on a globe. With such a
pitiful statistic, I decided to sample 284 people in a random sample, and thankfully, only 20 people
couldn’t find the US. Let p be the proportion of Americans who cannot find the US on the globe.
Round your answers to four decimal places.
A=90% confidence interval for p. Round your endpoints to four decimal places.
B=Standard deviation for the 1-proportion z-test on the alternative hypothesis: Ha:p<0.09.
C=Assuming p=0.09 is the correct parameter, the probability that it would take more than 9
people to find the first person who cannot find the US on a globe.
D=P-value of the 1-proportion z-test on the alternative hypothesis: Ha:p<0.09.
TBT Invitational 2013
Question #5
Statistics Team
Topic: Physics-Heisenberg’s Uncertainty principle states that it is impossible to know the position
and momentum of an electron instantaneously. In fact, the location of an electron around its
nucleus follows a probability distribution. The actual distribution is beyond the scope of AP
Statistics, but I can approximate the radial position of the hydrogen electron with the following
distribution, where r is the radial position of the electron, in Angstroms:
 80
3 r

 80
f  r    .5  r 
7
0


0  r  .15
.15  r  .5
else
Round your answers to four decimal places:
A=Probability that the electron stays within .15 Angstroms of the nucleus.
B=Median radial position of the electron.
C=Probability that the electron exists at r=.499 Angstroms
D=Probability that the electron stays within .10 and .20 Angstroms of the nucleus.
TBT Invitational 2013
Question #6
Statistics Team
Topic: English-The battle of vampires, wizards, and zombies persists in young adult fiction novels
today. I went to my local bookstore and found 23 books and searched for which mythical beings
existed. I found 11 books with vampires, 13 books with wizards, and 9 books with zombies. 3 books
had all three beings, 5 books only had vampires, 7 with just wizards, and 4 with only zombies.
A=the number of books with only vampires and wizards
B=the probability that a selected book would have two or more mythical beings.
C=the probability that a book has vampires, given that it has wizards
D=the probability that a book has zombies and wizards
TBT Invitational 2013
Question #7
Statistics Team
Topic: Lunchtime-Who doesn’t love tater tots? But don’t you feel like you never get enough of
them? For one lunch day, I kept track of the number of tater tots given out at Durant High
School and present them in the table, where X is the number of tater tots
X 10
P(X) 0.16
11
12
13
14
16
0.14
0.34
0.18
0.10
0.08
A=E[X]
B=Var[X]
C=Probability of receiving an odd number of tater tots
D= P  X  15 | X  12 
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Question #8
Statistics Team
Topic: History-Thomas Malthus stated that the means to survive would grow arithmetically while
the population would grow geometrically and we wouldn’t survive with enough resources. This got
you thinking about geometric and binomial distributions. Coming off of lunch, you think about a
basket of 20 apples with 7 red and 13 green apples. Round your answers to four decimal places.
A=Probability of pulling 5 apples at random and getting 2 red
B=Probability of pulling 7 apples and getting all 7 red
C=Probability of pulling your first green apple on the fifth pull (with replacement)
D=Probability that you pull at most 4 apples before getting your first red apple
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Question #9
Statistics Team
Topic: Geometry-A lattice point is a point in the coordinate plane with integer coordinates (i.e.
(1,1), (3,-4), and (0,1)). Consider the circle described by the equation x2+y2=9 inscribed in a square.
We will look at all lattice points inside and on the edges of the square.
A=Probability that a lattice point exists inside of the circle
B=Probability that a lattice point exists outside of the circle but inside the square
C=Probability that a point (not necessarily a lattice point) lies outside of the circle but inside the
square
D=Probability that a lattice point exists on the circumference of the circle, given that it lies on
the edges of the square
TBT Invitational 2013
Question #10
Statistics Team
Topic: Probability-Well, this is a statistics test, so I figure a classic probability question is a good
way to go! Let A and B be independent and exhaustive events with P  A   and P  B    . Answer
the following statements using α and β.
A= P  A  B 
B= P  A  Bc 
C= P  Ac  B c 
D= P  Ac 
TBT Invitational 2013
Question #11
Statistics Team
Topic: Biology-The Hardy-Weinberg equilibrium states that alleles (p and q=1-p) in a population
must exist in a certain ratio, but every person has two alleles. We survey 30 people (therefore 60
alleles) and find out that there are 12 q alleles present. Assume that the population parameter
frequency of the p allele is 0.84 and that all conditions for the sampling distribution are met.
A=The standard deviation of the p allele for this sample. Round your answer to 4 decimal places.
B=Proportion of subjects with a p and a q allele. The Hardy-Weinberg equilibrium states that this
must be equal to 2pq
C=Probability that the p allele frequency is less than 0.8 given our parameter frequency above.
Round your answer to four decimal places.
D=Using the empirical rule, the probability that the sample frequency of p is more than one
standard deviation from the mean.
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Question #12
Statistics Team
IQ is rather well characterized as a normal distribution with mean 100 and population standard
deviation 15. However, as the standard of living increases, the mean IQ score tends to increase.
Using this information, you randomly sampled 25 people and found that the mean IQ score was
107.20. You are curious to see if the mean IQ score has increased to 107. Assume a 5%
significance level, round off your critical z-score to three decimal places and round your answers
to four decimal places.
A=probability of committing a type I error
C=power of the test
TBT Invitational 2013
B=probability of committing a type II error
D=P-value of the test described in the problem
Question #13
Statistics Team
Let W={first 8 prime numbers}, X={first 8 composite numbers}, Y={first 8 even natural numbers}
and Z={first 8 Fibonacci numbers (a1=a2=1)}
A=the set with the largest mean
B=the set with the largest variance
C=the set with the largest maximum value
D=the set with the largest IQR
TBT Invitational 2013
Question #14
Statistics Team
Topic: Statistics-For the last question, give the term that corresponds to the following definition
or question about sampling methods and experimental design:
A=What sampling method is employed when the observational study samples every fifth person to
walk through a door?
B=An experiment is said to be this if neither the subject nor the experimenter know which
treatment is being given.
C=Only a well-controlled, replicable and randomized experiment can establish this between two
variables.
D=When sampling every house in one block of a large subdivision, what sampling method is
employed?
SOLUTIONS!!!!
STATS TEAM
TBT Feb 2013
1) All of this can be done via calculator!
A=0.9785
B=0.2045 C=6.1319 D=0.8333 (accept 5/6)
7.5  5 

2) A= P  z 
  P  z  1.25  0.1056
2 

P Y  0 
P  z  1
B= P Y  0 | Y  1 

 0.5142
P Y  1 P  z  0.5
C= P  X  6 | Y  6  P  X  6  P  z  0.5  0.3085 (since X and Y are independent)
D= P Y  X   P Y  X  0  P  Z  0 where Z=Y-X is normally distributed with mean 2-5=3 

3 and variance 4+22=8= 2 2 . Therefore, P  Z  0   P  z 
  0.1444
2 2

3) A= P  lithium  no relapse   0.166  0.107  0.273
B= P  relapse  imipramine   0.120  0.146  0.266
0.120
40

0.120  0.147 89
0.160 160
D= P  placebo|relapse  
Poorly worded also accept 160/227

0.513 513
4) A=(0.04545,0.0954) (via calculator)
C= P  relapse|only imipramine  
B=  p 
p0 1  p0 
n
0.09  0.91

284
 0.0170
C= P  X  9  1  0.09  0.4279
9
D=0.1245
5) The probability density function is two lines that meet at an apex at r=0.15 with a value of 4.
1
.15 4   0.3
2
B=Using above, find the height associated with the area of the trapezoid of 0.2:
1
80


0.2   m  0.15  4  .5  m   . Solving for m (graphically is best) gives two solutions:
2
7


.7958 and .2042. Only one of those is in the range: 0.2042
C=0
D=Split up this probability into two trapezoids:
1
80
 80
 1


P  0.1  r  0.2    0.05  .1  4    0.05  4   0.5  0.2    0.3524
2
7
 3
 2


A= P  r  0.15 
6) A=2
B=7/23
C=5/13
D=4/23
7) A= E  X   10  0.16   110.14   12 0.34  13 0.18  14 0.10  16 0.08   12.24
B=2.5824
C=0.14+0.18=0.32
D=0 (there is no X=15!)
2
3
 5   7   13 
8) A=       0.3364 also accept .3874
 2   20   20 
 7   7   13 
B=       0.0006 also accept 0
 7   20   20 
7
0
4
 7   13 
C=      0.0098
 20   20 
i 1
 13   7 
D=       0.8215
 20 
i 1  20 
4
9) Draw out the circle and square, you get
something like this: (dots not to scale, but they
their correct position)
A=25/49
B=0
36  9 4 - π
C=

36
4
D=4/24=1/6
10) A=α+β-αβ
B=α-αβ
C=1-αβ
D=1-α
11) A=
pˆ 1  pˆ 
.8(.2)

 0.0516
n
60
B= 2(.84)(.16)=0.2688 this should be p-hat.



C= P  pˆ  0.8   P  z 
0.8  0.84 
  0.1990
(.16)(.84) / 60 
D=1-0.68=0.32
12) A=0.05
1.645 
x  100
15 / 25
B= x  104.935
104.935  107 

P  IQ  104.935 on new H a curve=P  z 
  0.2456
3


C=1-0.2456=0.7544
D=0.0082
13) W={2,3,5,7,11,13,17,19} mean(W)=9.625, var(W)=40.8393, IQR(W)=11
are in
X={4,6,8,9,10,12,14,15} mean(X)=9.75, var(X)=14.5, IQR(X)=6
Y={1,1,2,3,5,8,13,21}
mean(Y)=6.75, var(Y)=49.9286, IQR(Y)=9
Z={2,4,6,8,10,12,14,16} mean(Z)=9, var(Z)=24, IQR(Z)=8
A=X
B=Y
C=Y
D=W
14) A=Systematic Sampling (or just Systematic)
B=Double blind
C=Causation (Causal link)
D=Cluster sampling (or just Cluster)