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Stat 571 First Midterm Exam October 9, 2007
Name:
- The exam is open book and open notes.
- Do all your work in the spaces provided. If you need additional space for your work, indicate clearly where the additional work can be found.
- The parts within a problem are not necessarily sequential.
- To receive full credit, you must show your work.
- Do not dwell too long on any one question. Answer as many questions as you can.
For instructor’s use:
1 27
2 6
3 15
4 15
5 10
6 27
Total 100
- (a) Preferred temperature is measured in a group of 10 blackbass fish, from a study about behavioral fever in fishes. For this sample, calculations yield
i yi^ = 295 and^
i y
2 i = 8708.66. Calculate the mean and variance of this sample.
(b) The histogram to the right represents preferred temperature data from bluegill sunfish (a different species of fish). How many fishes were there in this sample?
Make up a data set consistent with this histogram.
preferred temperature (C), bluegills
Frequency
27 28 29 30 31 32 33
0
1
2
3
4
(c) The stem and leaf plot to the right displays the data on pre- ferred temperature from fish after injection of a bacterial pyrogen. Make a boxplot for these data.
The decimal point is at the | 30 | 4 31 | 47 32 | 24 33 | 458 34 | 49 35 | 1 36 | 37 | 2
(d) The histogram to the right corresponds to the observed shift in preferred temperature after injection of the bacterial pyrogen. From visual inspection of the histogram, in this sample the mean is 1.6 2.6 3. the standard deviation is 0.6 1.6 2.
temperature shift (C)
Frequency
1.5 2.0 2.5 3.0 3.5 4.
0
1
2
3
4
- In a microarray experiment, suppose the differential expression of a particular gene follows the distribution shown to the right, which has mean 1.1 and standard deviation 1.05.
Now suppose differential expression of this particular gene is measured repeatedly from 30 independent microarrays, so that we get a random sample Y 1 ,... , Y 30 from the distribution shown to the right. Then the sample mean Y¯ has one of the distributions below: (a) (b) (c) (d) 0 1 2 3 4 differential expression
distribution density
Briefly justify your answer.
0.0 0.5 1.0 1.5 2.0 2.
(a)
0.0 0.5 1.0 1.5 2.0 2.
(b)
0.0 0.5 1.0 1.5 2.0 2.
(c)
0.0 0.5 1.0 1.5 2.0 2.
(d)
mean differential expression over 30 arrays (samples)
- In a study on the human odorant receptor OR7D4, 50 subjects with allele WM are asked whether they can detect the vanillin odor at a given (low) dilution. Assume here that the probability of detection is 12% in subjects with the WM allele. Calculate the probability that no fewer than 10 (i.e. 10 or more) of the 50 subjects detect the vanillin odor. If you make assumtions, justify them.
- A mother and father bear are catching salmon to feed themselves and their cubs. On a particular day, the mother bear will attempt to catch 4 salmon, and she has probability 0.7 to catch a salmon on each attempt. The father bear will attempt to catch 6 salmon, and has probability 0.6 of catching a salmon on each attempt. You may assume that every attempt is independent of every other attempt for both bears, and additionally that the mother’s attempts are independent of the father’s.
(a) Define a random variable Xm that is the total number of fish that the mother catches. What is the distribution of this random variable? If it is a random variable that we did not name in class, give its complete probability distribution (if discrete, all possible outcomes with the associated probabilities; if continuous, the probability density function). If it is a random variable that we did talk about in class, give its name and the values of all of the relevant parameters, and list the reasons why you think this random variable is of that type.
(b) Now define a different random variable Y that is the total number of fish that the mother and father catch combined. What is the expected value of this random variable?
(c) Suppose the mother and father bear must catch a combined total of at least 9 salmon in order for every member of the family to gain fat reserves for the coming winter. What is the probability that this occurs on a given day? Hint: it helps to enumerate the various ways of getting 9 salmon.
