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## Math.vu.nl

1. People who smoke die at a young age. Describe how you would collect data and which hypothesis you want to test to test this statement.
2. A researcher thinks that the Montignac diet doesn’t do anything. Describe how you would collect data and which hypothesis you want to test to test this statement.
3. An owner of a shoe shop wants to know whether advertisement helps. Describe how you would collect data and which hypothesis you want to test.
4. Males earn more money than females with the same education and age. Describe how you would collect data and which hypothesis you want to test to test this statement.
5. A person is tested for clairvoyance. He is shown the reverse of a randomly chosen play card 100 times and asked which suit it belongs to. The number of hits, or correct answers, is called X. As we try to find evidence of his clairvoyance, the null hypothesis is that the person is not clairvoyant. The alternative is: the person is (more or less) clairvoyant. We found x = 35. Test the null hypothesis.
6. A new vaccin is tested. 1000 volunteers get the vaccin and one month later they are infected with the virus. The vaccin is seen as a succes if 90% of the infected persons will (a) Formulate a statistical model and describe a test.
(b) Compute the rejection region. Take α0 = 0.05.
(c) Compute the power of the test if the probability a randomly selected volunteer does 7. A producer of cola is accused for putting too little cola into the bottles. The contents should be 500 ml. In total the contents of 12 bottles is measured. The measurements equal 500, 489, 501, 499, 489, 493, 499, 505, 493, 497, 500, 501. Test whether the bottles are not full enough at level 0.05. Assume that the contents of a random bottle has a normal distribution with unknown mean µ, but with known standard deviation 10.
8. Let X1, . . . , Xn be a sample from the N (µ, 4)-distribution. One wishes to test the null hypothesis H0 : µ ≤ 1 against H1 : µ > 1 at level α = 0.05. Compute the rejection region.
9. Let X1, . . . , Xn be a sample from the N (µ, 4)-distribution. One wishes to test the null hypothesis H0 : µ ≥ 1 against H1 : µ < 1 at level α = 0.05. Compute the rejection region.
10. In a factory 800 ton chemicals is produced a day. The production is measured for one week: 785, 805, 790, 793 en 802 ton is found. Test whether the production is too low (at level 0.05). Which assumptions do you make? 11. Ten sweaters are cut in two. One half is washed with washing-powder of brand A, the other half with washing-powder of brand B. Afterwards the length of the sweaters is measured (see table). Do sweaters washed with washing powder of brand A shrink less than sweaters 12. In an experiment the blood pressure of 32 individuals with hypertension is measured after taking the medicine Cozaar. In a second experiment the blood pressure of 20 individuals with hypertension is measured after taking the medicine Diovan. Both medicine decline the blood pressure. The values found in the two experiments are denoted as X1, . . . , X32 and Y1, . . . , Y20. The values found are: ¯ level 0.05) whether there is a difference between the effects of the medicines by computing 13. A producer of cola is accused for putting not enough cola into the bottles. The contents should be 500 ml. In total the contents of 12 bottles is measured. The measurements equal 500, 489, 501, 499, 489, 493, 499, 505, 493, 497, 500, 501. Test whether the bottles The producer gets a warning. The producer promisses to put more cola in the bottles.
Two months later he is checked again. The new measurements are 501, 499, 499, 497, 499, 515, 497, 497, 502, 501. Test whether the producer kept his promis (at level 0.05).
14. Ten pairs of chicks were selected to test the effect of a vitamin supplement on early growth.
The chicks in each pair were siblings of equal birth weight. One chick in each pair was given the supplement and the other was not. After two weeks, the weight of each chick was recorded. The researcher would like to test the research hypothesis that the supple- ment increases the growth rate of chicks in the first weeks after hatching against the null hypothesis that it has no effect. Describe an appropriate test.
15. Let X1, . . . , Xn be a sample from the N (µ, 4)-distribution. One wishes to test the null hypothesis H0 : µ ≤ 1 against H1 : µ > 1 at level α = 0.05. If µ = 2 the probability that the null hypothesis is rejected should be at least 0.9. Compute the minimal value for n if 16. Let X1, . . . , Xn be a sample from the N (µ, 4)-distribution. One wishes to test the null hypothesis H0 : µ ≥ 1 against H1 : µ < 1 at level α = 0.05. If µ = 0 the probability that the null hypothesis is rejected should be at least 0.9. Compute the minimal value for n if 17. In august 2005 3.5% of the adult Dutch population supported the political party D66. In august 2006 researchers want to test whether this percentage had changed. In order to do this they sampled 2500 adults. Of these 2500 adults 50 support the D66.
(c) Determine an approximated rejection region for X at level α = 0.05. Test the null hypothesis and give your conclusion.
(d) Compute the power in π = 0.025.
(e) The researches like to increase the power of the test. How can this be done? (f) If π = 0.025 the probability the null hypothesis is rejected should be at least 0.90.
18. In order to find out whether a coin is unbiased, the coin is tossed n times and the number of “head” comes up is counted. We want to test the null hypothesis H0 : π = 0.5 against H1 := 0.5, with π the probability “head” comes up. Take the level of the test equal to 5 %. What is the minimal sample size n to find a power of at least 0.9 in π = 0.6?

Source: http://www.math.vu.nl/sto/onderwijs/ks1voorMNW/TestingEnglish.pdf

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