Division of Mathematical Statistics

EXAM IN SF1900 PROBABILITY THEORY AND STATISTICS, WEDNESDAY OCTOBER 25, 2017 AT 08.00–13.00.

Examiner: Camilla Land´en, 08 – 790 61 97.

Means of assistance permitted: Numerical tables and compilation of formulae for this course, Mathematics Handbook (Beta) and pocket calculator.

You should define and explain your notation. Your computations and your line of reasoning should be written down so that they are easy to follow. Numerical values should be given with at least two significant digits. The number of exam problems is six. Each problems gives a maximum of ten points. Preliminarly, 24 points will guarantee a passing result. Students with 22-23 points on the exam will be given the right to complete their exam. Time and location for completion will be announced on the course web page. You must find out yourself if you are eligible for completion.

Points from the quiz exam and computer workshop during the first Autumn half semester will be accounted in accordance with the examination rules of the course.

The exam results will be announced at latest three working weeks after the day of the exam and will be retainable at the student affairs office during a period of seven weeks after the exam.

Problem 1

There are three possible types of error that can occur on a certain type of product: A, B and C, respectively. Errors of type A occur independently of errors of types B and C. Moreover, the probabilities that errors A, B and C occur are 0.05, 0.02 and 0.03, respectively. A product that has errors of type B also has errors of type C with probability 0.5. A product is considered to be defect if it has at least one of the error types. What is the probability that a randomly chosen

product is defect? (10 p)

Problem 2

The illumination system in a large reception hall consists of a large number of units that break down independently of each other. The time that elapses from the breakdown of one unit to the breakdown of the next unit is exponentially distributed with expectation 0.5 days.

a) Assume that 6 units have broken down. Determine the probability that it takes more than one day before another unit breaks down, that is until there are 7 broken units in the

illumination system. (2 p)

b) The company that is responsible for the illumination in the reception hall replaces broken units only after 100 units have broken down. Determine the probability that it takes more

than 45 days before broken units are replaced. (8 p)

Please turn over!

cont. exam in SF1900 2017-10-25 2

Problem 3

The survival function of a random variable X is defined as S_{X}(x) = 1 − P (X ≤ x). Let x_{1}, . . . , x_{n}
be a sample of the independent random variables X_{1}, . . . , X_{n}, all having the survival function
S_{X}(x) = 1 − x^{θ+1}, where 0 ≤ x ≤ 1 and θ > −1. Determine the least-square estimate of the

unknown parameter θ. (10 p)

Problem 4

A scientist at an Agricultural school wants to investigate if the content of quercetine, an antioxidant
considered to prevent cancer, differs between two kinds of apples. She measures the quercetine
content (in mg/100 g) in 8 apples of the first kind (x_{1}, . . . , x_{8}) and in 10 apples of the second kind
(y_{1}, . . . , y_{10}) and calculates x = ^{1}_{8}P8

i=1x_{i} = 4.4723, P8

i=1(x_{i}− x)^{2} = 0.1277, y = _{10}^{1} P10
i=1y_{i} =
4.6266, and P10

i=1(yi − y)^{2} = 0.1997. The measurements can be considered to be observations
of independent random variables X_{1}, . . . , X_{8} and Y_{1}, . . . , Y_{10}. It is assumed that X_{i} ∈ N (µ_{1}, σ),
i = 1, ..., 8, and Y_{i} ∈ N (µ_{2}, σ), i = 1, ..., 10, where the parameters µ_{1}, µ_{2} and σ are unknown.

a) Determine a 95% confidence interval for µ_{1} − µ_{2} and test on level of significance 5% the
hypothesis H_{0}: µ_{1} = µ_{2}. The conclusion should be clearly stated and motivated. (5 p)
b) Now assume, in contrast to in the (a)-part, that it is known from previous studies that
σ = 0.16. Determine a 95% confidence interval for µ_{1} − µ_{2} and test on level of significance
5% the hypothesis H_{0}: µ_{1} = µ_{2}. The conclusion should be clearly stated and motivated. (5 p)

Problem 5

Sm˚att & gott is a chocolate sweet consisting of milk chocolate draped with coloured caramel. The company that manifactures the sweets claims that the occurence of the different colours are as in the table below:

Brown Yellow Red Orange Blue Green

30% 20% 20% 10% 10% 10%

A large bag of Sm˚att & gott-sweets was randomly chosen and turned out to contain 58 brown, 45 yellow, 27 red, 14 orange, 12 green and 13 blue sweets. Perform an appropriate test to determine if this result supports the claim of the company. State your hypotheses and motivate your conclusions

carefully. Use significance level 5%. (10 p)

Problem 6

An astronomer in Sweden wants to measure the distance µ to a bright star. She carries out n_{1} me-
asurements, x_{1}, . . . , x_{n}_{1}, considered to be samples of independent random variables X_{1}, . . . , X_{n}_{1}. A
colleague in Chile wants to measure the same distance and conducts n_{2} measurements, y_{1}, . . . , y_{n}_{2},
considered to be samples of independent random variables Y_{1}, . . . , Y_{n}_{2}. The measurements of the
Swedish instrument have standard deviation σ_{1} and the measurements of the Chilean instrument
have standard deviation σ_{2}. None of the instruments are biased. The astronomers want to combine
their estimates of the distance, ¯x and ¯y, using the linear combination µ^{∗}_{obs} = c_{1}· ¯x + c_{2}· ¯y, where c_{1}
and c_{2} are two constants. State the condition on the constants c_{1} and c_{2} that makes the estimate
µ^{∗} unbiased. Assume further that n_{1} = n_{2} and that σ_{1} = 2 · σ_{2}. What values of the constants c_{1}
and c_{2} gives the most efficient estimate µ^{∗}_{obs} = c_{1}· ¯x + c_{2}· ¯y? (10 p)

Good luck!