1. 15/06/2015Dr Andy Brooks1 TFV0103 Tölfræði og fræðileg vinnubrögð Fyrirlestur 13 Yfirlit og Spurningar (tölfræði)

2. 15/06/2015Dr Andy Brooks2 Leiðbeiningar í lokaprófinu TFV0103 Tölfræði 1. Svarið öllum spurningum. 2. Setjið hring við einn valmöguleika, a), b, c) eða d). 3. Leyfilegt er að nota reiknivél. 4. Leyfilegt að nota formúlu blaðið úr kennslubókinni með (i) handskrifaða þýðingu á enska textanum og (ii) handskrifað formulas. 5. Það er ekki leyfilegt að nota kennslubókina eða glósur. 6. Prófið er bæði á ensku og á íslensku. Vinsamlegast svaraðu bara íslensk spurningar. 1. Answer all the questions. 2. Answer by circling one of a), b, c) or d). 3. You have permission to use a calculator. 4. You have permission to use the formula sheet from the textbook on which may appear (i) handwritten icelandic translations of the english text and (ii) handwritten formulae. 5. You do not have permission to use textbooks or notes. 6. The exam is given in both english and icelandic. Please only answer the icelandic questions.

3. 15/06/2015Dr Andy Brooks3 Reiknivél Ertu búin(n) að velja? Kannt þú vel að nota? Komdu með rafhlaða til viðbótar

4. 15/06/2015Dr Andy Brooks4 Lokapróf Krossaspurningar eins og heimapróf Ekki alveg eins – tölur breytast til dæmis. 30 spurningar Kafli 1,2,3 – 4 spurningar Kafli 4,5,6,7,8,9 – 3 spurningar BOOKTABLE 3 og 6 eru með prófblöðin Þú verður að koma með formúlu blaðið

5. 15/06/2015Dr Andy Brooks5 Two kinds of statistics Descriptive statistics: collecting, describing and presenting sample data. Inferential statistics: drawing conclusions about populations. A population/þýði is a collection of individuals or objects or events whose properties are to be analyzed. A sample/úrtak is a subset/hlutmengi of the population. Iceland has a population of 290,000 but how many Icelanders responded to the question about the 5 kr. coin? 10 eða 100 eða 10.000? Alltaf, þu verður að vita af úrtaksstærð. lýsandi tölfræði ályktunartölfræði F2

6. 15/06/2015Dr Andy Brooks6 Types of variables Variable Quantitative Qualitative Discrete Continuous Nominal Ordinal red, blonde,brown,black hot, warm, cool, cold weight, height age in years, number of modules hot>warm>cool>cold cannot calculate averages megindleg eigindleg rofin samfelld nafnbreyta raðbreyta breyta Þyngd, hæð aldur, hve mörg námskeiðar F2

7. 15/06/2015Dr Andy Brooks7 Graphing Qualitative Data Types of 498 hospital operations/aðgerðir eigindarleg gögn Ta02-01 Pie Chart/Skífurit Bar Graph/Súlurit Hvaða graf er best? space/rúm between bars F3

8. 15/06/2015Dr Andy Brooks8 The sample variance, s 2, is the mean of the squared deviations, calculated using n  1 as the divisor: Sample Variance (s 2 ) & Standard Deviation (s) s n xx 22 1 1     () where n is the sample size Andy says: try to remember these formulae. dreifni staðalfrávik F3

9. 15/06/2015Dr Andy Brooks9 Positive and negative correlations r = 0,71 positive correlation/ r = -0,74 negative correlation neikvæð fylgnijákvæð fylgni \misc\movies\JK_31 Interactivity 3A F4

10. 15/06/2015Dr Andy Brooks10 Að reikna út Pearson´s r Pearson fylgnistuðull x y SS sum of squares/ summa fervika F4

11. 15/06/2015Dr Andy Brooks11 Regression line/Jafna bestu línu xy 35 48 56 67 79 86 910 11 n=8 (x,y) 3,5174 F5

12. 15/06/2015 Að reikna út b 1 og b 0 xyx2x2 xy 35915 481632 562530 673642 794963 866448 9108190 1011100110 ∑x∑ y∑x 2 ∑ xy 5262380430 n=8 ekki 100% rétt - ekki nógir tugastafir að reikna með F5

13. 15/06/2015(c) Thomson Learning Inc.13 Experimental results of tossing a coin 10 times each trial: relative frequency/hlutfallsleg tíðni F6

14. 15/06/2015Dr Andy Brooks14 Dæmi Kasta pening fimm sinnum What is the probability of: A = {exactly one head/nákvæmlega einn fiskur}? B = {exactly five heads/nákvæmlega fimm fiskar}? 32 possible outcomes/32 möguleikir (úrtaksrúmið) 2 x 2 x 2 x 2 x 2 A={HTTTT,THTTT,TTHTT,TTTHT,TTTTH} B={HHHHH} TTHHH F6

15. 15/06/2015Dr Andy Brooks15 General Addition Rule Let A and B be two events defined in a sample space/úrtsaksrúm S: If two events A and B are “mutually exclusive”: P(A and B) = 0. rule/regla sniðmengi er ekki til Intersection/Sniðmengi - við ætlum ekki að telja tvisvar. F7

16. 15/06/2015Dr Andy Brooks16 Space shuttle Challenger disaster Reliability of a single joint was 0.9777. But the reliability of six joints was: 0.9777 x 0.9777 x 0.9777 x 0.9777 x 0,9777 x 0.9777 eða 0.869 u.þ.b. 87% Video: Chapter 4 and 5 probability reliability/áreiðanleiki Special Multiplication Rule independent/óháður F7

17. 15/06/2015Dr Andy Brooks17 Binomial Distribution/Tvíkostadreifing p is the probability of “success” on a single trial –t.d. “probability of heads” P(H) = 0,5 so p = 0,5 q is the probability of “failure” on a single trial –t.d. “probability of tails” P(T) = 0,5, so q =0,5 P(x), the probability there will be exactly x successes in n trials, is: binomial coefficient/stuðull factorial success/velgengni failure/bilun F8

18. 15/06/2015Dr Andy Brooks18 {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} n = 3 (independent trials) “success” = H og “fail” = T p = P(H) = 0,5 og q = P(T) = 0,5 p + q = 1 random variable x = 0, 1, 2 eða 3 independent/óháður random variable/slembibreyta F8

19. 15/06/2015Dr Andy Brooks19 Interactivity 6A 1,00,5 ≈ 0,95 (2σ) ≈ 0,68 (1σ) area/flatarmál F9

20. 15/06/2015(c) Thomson Learning, Inc.20 Dæmi What is the area under the standard normal distribution between z = 0 and z = 1.45 ? Table 3 z0.000.010.020.030.040.050.06 1.40.4265............ z-gildi Number of standard deviations from the mean. Hve mörg staðalfrávik frá meðaltalinu. z has a standard normal distribution. US. flatarmálið undir feril frá,til hvert er flatarmálið milli z = ? og ? F9

21. 15/06/2015Dr Andy Brooks21 RVLS Sampling Distributions σ=5,00 S=1,02 F10

22. 15/06/2015Dr Andy Brooks22 Standard error of the mean/Staðalvilla The standard deviation of the sampling distribution of sample means is called the standard error of the mean. F10

23. 15/06/2015Dr Andy Brooks23 Confidence Interval (CI)/Öryggisbil Oftast, við erum með eitt úrtak og að reikna út meðaltal úrtaks. Úrtaksmeðaltal er punktspá fyrir þýðismeðaltal. 95% öryggisbil fyrir μ (α=0,05) er: 99% öryggisbil fyrir μ (α=0,01) er: Ef σ er þekkt. point estimate/punktspá F11

24. 15/06/2015Dr Andy Brooks24 Dæmi The consumer group take a sample of 40 pills (úrtakstærð n=40) and find: The test statistic z* is: The probability z<z*, P(z<-1,58), from BOOKTABLE3, is 0,5 - 0,4429 = 0,0571 F11

25. 15/06/2015Dr Andy Brooks25 Student´s t-statistic Because s is an estimate, instead of the test statistic z* and BOOKTABLE3, we use the Student´s t statistic: The t-distribution is not normally distributed for small sample sizes n. The t-distribution depends on the “degrees of freedom” df = n-1 (for the examples in Chapter 9). degrees of freedom/frígráðurt-distribution/t-dreifing F12

26. 15/06/2015Dr Andy Brooks26 Women´s heights/Hæðir kvenna s er punktspá fyrir σ 95% öryggisbil er: Áður, notuðum við 1,96 áður ≈ 4mm meira F12