Download Quiz#2 Key

Biometry (BIOL4090) Quiz #2.
Student name: _______________ KEY _______________
This 30-minute quiz is worth 5 points. Show all your work to get partial (full) credit. You may use a
calculator, but not a smart phone. You may also leave calculations as ratios if necessary. I have extra
paper, if you need some. Write your name on every page and staple them together with this cover page.
1) Define the following terms, making sure you include the terms in parenthesis (+0.25 each):
p-value (null hypothesis): The probability of sampling the observed data (by chance), if the null
hypothesis is, in fact, true
alpha (type-I error): due to the mechanics of hypothesis testing, alpha (usually set at 0.05)
defines the threshold for statistical significance and the probability of committing a type-I error
critical value (alpha): The minimum value that a test statistic must have to produce a
significant result; when the p value equals the alpha level
conditional probability (independent): Defined as the lack of independence (or the dependence) in
multiple events, such that the probability of one event occurring is influenced by whether another
event happened. Note: Two events, A and B, are independent if the fact that A occurs does not affect
the probability of B occurring
2) Draw a diagram showing the four possible outcomes when we test a null hypothesis. On that 2 by 2
grid, make sure you label the following cells: false positive, false negative, type-I error, type-II error,
error of falsity, error of ignorance (+0.25 each). Please use the space provided below.
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3) Draw a diagram of the hypothetico-deductive method, using two alternate hypotheses (+0.50)
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4) You are studying a caterpillar-wasp
parasite-host system. The wasps lay eggs on
the caterpillars, which develop by eating
their hosts. Wasps can determine the sex of
their offspring, and do so when they lay the
single egg in the caterpillar. While wasps
can only lay one egg at a time, multiple
wasps can lay eggs into the same caterpillar.
You want to test the hypothesis that the sex
of the egg is independent from the number of
eggs laid in a caterpillar (1 or 2). This
following probability tree summarizes your
observations.
Briefly explain how can you determine whether these two events (presence of an egg in the caterpillar
AND the sex of the egg being laid next) are independent (+0.25) ?
Conceptually, if two events are independent, the probability that one occurs is not influenced by
whether the other one does or does not occur. In this case: the first event is whether the egg was laid
already in the caterpillar and the second event is the sex of the second egg laid on the caterpillar.
Practically, show your calculations for determining whether the two events described above are, in fact,
independent from each other (+0.25 for the male egg and +0.25 for the female egg).
So, you need to show that the probability of laying a male or a female egg do not depend on whether an
egg was already laid in the caterpillar.
P (male) = 0.36. It can happen in two ways:
P (no egg already laid, male egg) = 0.8 * 0.3 = 0.24
P (one egg already laid, male egg) = 0.2 * 0.6 = 0.12
P (female) = 0.64. It can happen in two ways:
P (no egg already laid, female egg) = 0.8 * 0.7 = 0.56
P (one egg already laid, female egg) = 0.2 * 0.4 = 0.08
THE TWO EVENTS ARE NOT INDEPENDENT because if the probability of the complex event is
different from the product of the probabilities of the two single events:
P(egg already laid) = 0.2 AND P(male) = 0.36, but P(male and egg already laid) = 0.12
(NOT 0.072)
P(egg already laid) = 0.2 AND P(female) = 0.64, but P(female and egg already laid) = 0.08
(NOT 0.128)
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5) There are two species of whales, with different relative abundances, randomly distributed across the
North Pacific: 7500 white whales and 2500 black whales. No matter what the actual whale species
they see, observers get the ID right 80% of the time and make a mistake with the ID 20% of the time.
Draw the probability tree describing this scenario (+0.75). Make sure you indicate the probabilities for
each node and for each possible path along the tree. Hint: start with the whale abundances (as 1st event)
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Answer the following questions, relating to the previous probability tree.
If you go on a cruise and see a whale: what is the probability it is a black whale? (+0.25):
(show your work)
p (black whale) = 2500 / (7500 + 2500) = 0.25
If you go on a cruise and see a whale: what is the probability it is a white whale? (+0.25):
(show your work)
p (white whale) = 7500 / (7500 + 2500) = 0.75
OR
1 – p (white whale) = 1 – 0.25 = 0.75
If you identify the whale as a black whale, what is the probability it really is a black whale? (+0.25):
(show your work)
Given the conditional probability: the observer identified the whale as a black whale.
P (black whale & ID as black) = 0.20 (from probability tree)
P (ID as black) = P (black whale & ID as black) + P (white whale & ID as black) = 0.20 + 0.15
P (black whale given that ID as black) = 0.20 / (0.20 + 0.15) = 0.57
If you identify the whale as a black whale, what is the probability it really is a white whale? (+0.25):
(show your work)
Given the conditional probability: the observer identified the whale as a black whale.
P (white whale & ID as black) = 0.15 (from probability tree)
P (ID as black) = P (black whale & ID as black) + P (white whale & ID as black) = 0.20 + 0.15
P (black whale given that ID as black) = 0.15 / (0.20 + 0.15) = 0.43
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