STAT C141, Spring 2005
HW#3; Due 03/17/05
Problem 1
Consider the simple random walk with h=0, b=1 and a=-L, where L is positive. Use equation with (for details, see lecture notes on “Probability theories in simple random walk”) to write down the probability that the walk eventually reaches 1 rather –L. For the case pq, show that the limiting value of this probability as is .
Problem 2
The limiting () probability found in Q1 is the probability in an unrestricted random walk which starts at 0 and has pq, that the walk ever reaches +1. Show this implies that in the unrestricted case, the probability that the unrestricted walk ever reaches the value y is , for any positive integer y.
Problem 3
Several thousand measurements on a check weight average out to 512 micrograms above a kilogram and their SD is 50 micrograms. Then the weight is cleaned. The next 100 measurements average out to 508 micrograms above one kilogram and their SD is 52 micrograms. Apparently, the weight got 4 micrograms lighter. Or is this chance variation?
a)Would you estimate the SD of the box as 50 or 52 micrograms?
b)Would you make a z-test or a t-test?
c)Did the weight get lighter? If so, by how much?
Problem 4
National data shows that on average, college freshmen spend 7.5 hours a week going to parties. One administrator does not believe that these figures apply at his college, which has nearly 3,000 freshmen. He takes a simple random sample of 100 freshmen and interviews them. On average, they report 6.6 hours per week going to parties and the SD is 9 hours. Is the difference between 6.6 and 7.5 real?
Problem 5
Breast-feeding infants for the first few months after their birth is considered to be better for their health than bottle feeding. According to several observational studies, withholding the bottle in hospital nurseries increases the likelihood that mothers will continue to breast-feed after leaving the hospital. As a result, withholding supplementation has been recommended. A controlled experiment was done by K.Gray-Donald, M.S. Kramer, and associates at the Royal Victoria Hospital in Montreal. There were two nurseries. In the "traditional" one, supplemental feedings of newborn infants were figen as usual: a bottle at 2 a.m., and whenever the infant seemed hungry after breastfeeding. In the experimental one, mothers were awakened at 2 a.m. and asked to breast-feed their babies; supplemental feeding was discouraged. Over the four-month period of the experiment, 393 mothers and their infants were assigned at random to the traditional nursery, and 388 to the experimental one. The typical stay in the hospital was 4 days, and there was followup for 9 weeks after release from the hospital.
a)At the end of 9 weeks, 54.7% of the mothers who had been assigned to the traditional nursery were still breast-feeding their infants, compared to 54.1% in the experimental nursery. Is this difference statistically significant? Formulate the Null and the alternative hypothesis, and perform a test on a significance level of 5%.
b)It was really up to the mothers whether to breast-feed or bottle-feed. Were their decisions changed by the treatments? To answer that question, the investigators looked at the amounts of bottle-feeding in the two nurseries, expressed a milliliters per day (ml/day). In the traditional nursery, this averaged 36.6% ml/day per infant, and the SD was 44.3; in the experimental nursery, the figures were 15.7 and 43.6. What do you conclude?
c)Did the different treatments in the two nurseries affect the infants in any way? To answer that question, the investigators looked at the weight lost by each infant during the stay, expressed as a percentage of birth weight. In the traditional nursery, this averaged 5.1% and the SD was 2.0%; in the experimental nursery, the average was 6.0% and the SD was 2.0%. What do you conclude? (It may be surprising, but most newborns lose a bit of weight during the first few days of life.)
d)Was the randomization successful? To find out, the investigators looked at the birth weights themselves (among other variables). In the traditional nursery, these averaged 3,486 grams and the SD was 438 grams. In the experimental nursery, the average was 3,459 grams and the SD was 434 grams. What do you conclude?
Problem 6 Searching SWISS-PORT
In this assignment, you will send a query sequence to a BLAST server, which find the “best” gapped local alignment between your query sequence and a large number of sequences in the protein database SWISS-PORT. . Alignment will be scored using the BLOSUM62 scoring matrix.
(a)Please do the following:
1)Go to the NCBI BLAST page and click on “Protein-protein BLAST (blastp)” which is under “Protein” section. Then you enter the page of “protein-protein BLAST”.
2)Paste the query sequence f8i2.fasta (the file can be downloaded from class webpage) to ‘search’ box. Here f8i2.fasta is an abbreviation for your query sequence, human factor VIII intron 22 protein, in fasta format. F8I2_HUMAN is the Swiss-port name for your query sequence.
3)Choose “Swissport” in the ‘choose database’ box. Now click on “BLAST!” You will enter the page of “formatting BLAST”.
4)Click on “Format!” in the page of “formatting BLAST”. Now you enter the results sheet.
(b)Repeat (a) with query sequence f8i2_perm.fasta, a scrambled version of f8i2. This sequence has the same amino acid composition as f8i2, but (with very high probability) should not achieve any meaning matches.
(c)Please turn in the results sheets from (a) and (b),