Avd. Matematisk statistik
EXAM IN SF1901 PROBABILITY THEORY AND STATISTICS MONDAY JANUARY 9, 2017, 2PM–7PM.
Examiner: Thomas ¨Onskog, 08 – 790 84 55.
Permitted aids: List of formulae and tables in mathematical statistics, Mathematics Handbook (Beta), quick reference for the calculator, calculator.
Notation must be explained and defined. The calculations and reasoning must be so explicit that they can be easily followed. Numerical answers must be given with at least two valid digits. The exam consists of 6 problems. A correct solution of a problem yields 10 points. Students with at least 24 points are guaranteed to pass. Students with 22–23 points will have the opportunity to complement the exam. The time and place of the supplementary exam will be announced on the course web page. You have to find out for yourself if you are entitled to take the supplementary exam.
Bonus points from the quiz and the computer exercises given in period 2, Autumn 2016, will be added to the final result of this exam, given that you obtain at least 20 points at this exam.
The exam will be corrected within three working weeks from the date of the exam, and will then be available at the student expedition for at least seven weeks after the exam.
Uppgift 1
In biathlon, contestants ski around a track stopping at each of four shooting stations. At each shooting station the contestant attempts to hit five targets. The contestant may use up to eight shots, however the three last shots must be loaded one at a time. If despite eight shots spent, the contestant has not hit all five targets, the contestant must ski penalty laps, one for each missed target.
Given that a certain contestant hits a target with a probability of 0.7, calculate the probability that no more than two out of the contestant’s four shootings result in penalty laps. Hits are
considered independent of one another. (10 p)
Uppgift 2
Women have two X-chromosomes, while men have one X-chromosome and one Y-chromosome.
A child will inherit at random an X-chromosome from its mother, and randomly either an X- chromosome or a Y-chromosome from its father. The sex of the child is determined by which chromosome it inherits from its father. This exercise considers color blindness, which is recessively inherited via the X-chromosome. A woman is color blind if both X-chromosomes carry the gene variant for color blindness. A man is color blind if his single X-chromosome carries the gene variant. Let K denote a non color blind woman. Assume that the probability that one of K’s X-chromosomes carries the gene variant for color blindness is 1/3, and that the probability that none of K’s X-chromosomes carry the gene variant for color blindness is 2/3.
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a) Compute the probability that a child of K’s does not inherit the gene variant for color
blindness from its mother. (4 p)
b) Suppose that K gives birth to a son, and that the father of the son is not color blind.
Compute the probability that one of K’s X-chromosomes carries the gene variant for color
blindness given that the son is not color blind. (6 p)
Uppgift 3
The enrichment (in %) in twelve of the fuel rods of a nuclear reactor has been measured resulting in the following data:
2.94 2.75 2.95 2.81 2.95 2.90 2.82 2.95 3.00 2.95 3.00 3.05
Based on these data, denoted by x1, . . . , x12, it is claimed that the expected value of the enrichment is 2.95%.
a) Find a confidence interval with confidence level 90% for the expected enrichment. As sta- tistical model you should use that the measurements are a random sample from N(µ, σ) (Xi ∈ N(µ, σ), i = 1, . . . , 12).
Computational help: P12
i=1(xi− x)2 = 0.085025, x = 121 P12
i=1xi = 2.9225. (7 p) b) Test the null hypothesis
H0 : µ = 2.95 against the alternative hypothesis
H1 : µ 6= 2.95
at significance level 10%. Your conclusion about whether to reject H0, or not should be
clearly stated and motivated. (3 p)
Uppgift 4
The party preference survey of Statistics Sweden, SCB, is conducted twice a year and it is the biggest of its kind in Sweden. Slightly simplified a nation wide random sample of persons who are eligible to vote in the Riksdag election is asked to answer the question “Which party would you vote for if an election were to be held in the next few days?”. Below you will find the results from the surveys conducted in May and November 2016 for the two largest parties in the Riksdag.
Party May November Change
Socialdemokraterna 29.5% 29.2% −0.3%
Moderaterna 24.7% 22.8% −1.9%
The total numbers of answers received was 5021 in November and 4838 in May.
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a) Find confidence intervals with approximate confidence level 95% for the changes in the proportion of voters claiming that they would vote for “Socialdemokraterna” and “Modera-
terna”, respectively (one interval for each party). (7 p)
b) Test whether the change in the proportion of voters for “Socialdemokraterna” is significant at the approximate level 5%. Test whether the change in the proportion of voters for “Mo- deraterna” is significant at the approximate level 5%. Carefully state your hypotheses and
conclusions. (3 p)
Uppgift 5
The Danish missionary Hans Egede Saabye who was stationed in Greenland during the years 1770 to 1778 wrote in his diary the following: ”In Greenland, all winters are severe, yet they are not alike. The Danes have noticed that when the winter in Denmark was severe, as we perceive it, the winter in Greenland in its manner was mild, and conversely”. In order to test the claim that there is a relationship between the winter temperatures in Greenland and in Denmark one could use the table found below. The table is based on measurements of the average temperature in January in Nuuk in Greenland and in the Danish capital Copenhagen during the years 1866 to 2013. For each year and city the month of January has been placed in one out of three categories: severe (if the average temperature is a least 0.8 standard deviations below the normal value), mild (if the average temperature is a least 0.8 standard deviations above the normal value), or normal (if the average temperature deviates less than 0.8 standard deviations from the normal value).
Severe winter Nuuk Normal winter Nuuk Mild winter Nuuk
Severe winter Copenhagen 2 19 9
Normal winter Copenhagen 13 51 22
Mild winter Copenhagen 12 18 2
Test at the 1% risk level whether there is a dependence between the winter temperature in Gre- enland and the winter temperature in Denmark, or not. Make sure to state your hypotheses and
conclusions. (10 p)
Uppgift 6
In an electrical component uniformly distributed noise X ∈ U (0, 1) is transformed monomially according to
Y = Xγ, where the degree γ > 0 is a constant.
a) Determine the density function of the transformed noise Y . (4 p) b) Let y1, ..., yn be n independent observations of the transformed noise. Find the ML estimate
of γ and determine whether it is unbiased or not. (6 p)
Good luck!
