Course Notes - Department of Mathematics and Statistics

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Gamebird survey results4 May 25 May 15 Jun 29 JunEstimate SE Estimate SE Estimate SE Estimate SEMallards harvestedR 6.17 0.58 4.36 0.67 2.80 0.51 2.38 0.41N 5.23 0.75 4.02 0.62 1.74 0.44 1.24 0.34Hours huntedR 9.67 0.57 8.18 1.16 5.43 0.76 4.91 0.65N 6.85 0.59 5.66 0.81 3.16 0.61 1.71 0.34Harvest by respondents (R) and non-respondents (N) to gamebird hunting questionnaire.Example 2: 1936 Presidential election• Literary Digest predicted 2:1 victory to Landon (R) over Roosevelt(D).• Election: Landon 2 states Roosevelt 46!• Mailout to more than 10 million – 2 million responded• Frame: readers of Literary Digest, telephone numbers, registeredcar owners.• George Gallup used a quota sample of 50,000 people to (a) predicta win for Roosevelt and (b) used a sample of 3000 from theLiterary Digest frame to predict that Literary Digest would mispredict.Probability sampling• We want our frame to match the population of interest and a wayto draw a representative sample.• Probability sampling is the only way to ensure representativeness• Simple random sample For a finite population of size N drawa sample of size n such that each possible sample has the sameprobability.- Lotto - each draw of 6 balls has the same probability18
- 1 in 3.8 million - 3.8 million distinct sequences, all equallylikely.- Sampling without replacementSimple random sample• Easy to analyse• let y 1 , . . . , y n denote the observed valuesEstimate population mean:∑ ni=1ˆµ = ȳ =y i.n‘add the values up and divide by their number’.Estimate population total:ˆT = N nn∑y i = Nȳ.‘take the sample mean and multiply by the population size.’i=1Stratified sampling• Much of statistical design theory is about controlling variation.• More variable data means less precise inference (signal vs noise).• Stratified sampling useful when the population comprises differenttypes of similar individuals.- A stratum is a population sub-division of similar units• Take a simple random sample from within each stratum.- More precise for the same expenditure.- Might be interested in the results by stratum.• Can take different sized samples from different strata.- Device for reducing the overall variability.• Formulae for estimates more complicated.- Consult an expert.19
Page 1 and 2: Contents1 Course Administration 42
Page 3 and 4: 10 Regression 20210.1 Introduction
Page 5 and 6: Resource PageDepartment of Mathemat
Page 9 and 10: 3 Data and Study Designs3.1 Basic d
Page 11 and 12: Parameter• Parameter: Fixed numbe
Page 13 and 14: TypesDiscrete - can put in one-to-o
Page 15 and 16: - How do geologists know?- Sediment
Page 17: Source: skyscrapercity.comSampling
Page 23 and 24: Important designs• Completely ran
Page 25 and 26: B. Randomised control - of all chil
Page 27 and 28: - Reason why observational studies
Page 29 and 30: Things that go wrong• Even the be
Page 31 and 32: 4 ProbabilityFred’s DayFred awoke
Page 33 and 34: • The event that the mouse we tra
Page 35 and 36: Blood donor example - Multiplicatio
Page 37 and 38: Fair Die Example• A fair die is t
Page 39 and 40: Tree diagram rules• Add Verticall
Page 41 and 42: Tree Diagrams - Independent Stages0
Page 43 and 44: Dependent Stages• Andrew, John an
Page 45 and 46: 0.90BBiopsy +ve (true positive)0.00
Page 47 and 48: Calculating Probabilities• Estima
Page 49 and 50: 4.3 Random VariablesRandom Variable
Page 51 and 52: Calculating the Variance• The sam
Page 53 and 54: Examples• Consider a data set XX
Page 55 and 56: Combining 2 Random Variables• If
Page 57 and 58: Fred’s DayShould Fred be worried
Page 59 and 60: Now using the complementary event r
Page 61 and 62: Mean of Binary DistributionThe mean
Page 63 and 64: • Mean number of successes• Var
Page 65 and 66: Using the Formulan = 3, x = 2, π =
Page 67 and 68: • A probability less than 0.05 is
Page 69 and 70:
Endangered bird egg example solutio
Page 71 and 72:
RELATIVE FREQUENCY HISTOGRAMNormal
Page 73 and 74:
Finding Areas Under the Curve• In
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• Find P r(−1 < Z < 1.64)pnorm(
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INVERSE PROBLEMS USING R-COMMANDER
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CALCULATING PROBABILITIESAssume tha
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CONTINUITY CORRECTION• Normal pro
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Sample size = 10Consider the situat
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ConclusionWhat conclusion would you
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10%5.5 6.0 6.5 7.0 7.5 8.0 8.5XWe k
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6 Sampling Distributions and Estima
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DERIVATION IWe can think of the sam
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Solution IIPr(X > 172) = 1 - pnorm(
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qnorm(0.975) as 2.5% of the area is
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6.2.1 Sample Size CalculationEXAMPL
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The 99% Confidence IntervalThe 99%
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6.3 Comparing Two SamplesCOMPARING
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THE POOLED VARIANCEIf the variances
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SOLUTION - Calculating the confiden
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Confidence Interval for raw dataTo
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= 6.6 ± 24.6That is, -18.0 < µ in
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STANDARD DEVIATIONIf P = X n : σP
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6.4.2 Sample Size CalculationEXAMPL
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Fred for MayorFred has decided to t
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Fred’s mate Bob pointed out that
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ALTERNATIVE HYPOTHESESTwo types:Stu
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• If we use the sample standard d
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Calculate the test statistic:Test s
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Finding the pooled variance:Recall
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Caluclate the estimated standard er
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7.6 Significance and Conclusiveness
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Conclusion: There is no evidence th
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Some practical pointers• Aim for
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Fred’s Parking & Extracurricular
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Fred calculated the p-value using t
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Fred told Carol he would use odds r
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Flu vaccine example• There were 1
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8.3 Attributable Risk (AR)The attri
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• Therefore w x ≈ww+x and y z
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Aspirin study example - gastrointes
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Mobile phone example• Human expos
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OR = 0.81 with CI (0.39,1.70)• p
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• We calculate these expected cou
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• We can calculate the p value us
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χ 2 =(100 − 108.99)2 (107 − 11
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• The reason for the discrepancy
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Summary - Confidence IntervalsFacto
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Since 1 is excluded from this 95% i
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9 ANOVAFred’s Public ImageSince F
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• Previously used two sample t-te
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• This tells us if the observed d
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RESIDUAL MEAN SQUARE (s 2 e)Group A
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9.2 Post ANOVA AnalysisPOST ANOVA A
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• Hence no need to additionally c
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Since zero is excluded, and the int
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SUM OF SQUARESTotal SS = 93 2 + 100
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General Level SS = 12 x ( 9812 )2=
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ANOVA TABLEA one factor analysis of
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Data > Manage variables in active d
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DATA COMPONENTSEach data value can
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ANOVA TABLE - Two Factor FactorialS
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A B C D EChristchurch 120.5 119 122
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DataFERTILISER LEVELSEED TYPE Low M
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Interaction PlotsHypothesis test fo
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Fred’s WeightUsing the informatio
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Fred wanted to address the followin
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10 RegressionFred’s Dog/BeardFred
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2. studies which have measured bina
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• What if we colour code the poin
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Statistically• We use slightly di
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Blood pressure exampleStress(x) Blo
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Focussed look on the limits of our
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Height example• We will perform a
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Normality Assumption FAILPP PlotExp
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10.3 Confidence Intervals and Regre
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0.236 < β 1 < 0.496• We can conc
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95% confidence and prediction inter
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100m World Record Progression●●
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Negative correlation• The denomin
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Stress exampleStress(x) Blood Press
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10.5 Multiple regression• Simple
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• Appears that total lung capacit
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Multiple linear regression predicti
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Three variable modelIntepreting Mod
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• So for men we have:y = −7.066
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• Is the variable age an importan
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Extra sum of squares principle• T
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Diagnostics• Remember our Assumpt
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TableTreatmentControlBP(Y) AGE(X) B
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• So for people in the control gr
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Discussion• The value of d is the
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Multiple Linear RegressionTreatment
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• Logistic regression is sometime
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• Given the s.e.(OR) = 0.751. Thi
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• We get the same as when we calc
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Female(0)Male(1)Born Left(1) Right(
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Fred’s Dog/BeardTo help solve his
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This is a very high r 2 value and i
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Attractiveness = 4.85 − 0.03 × (
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• Because the calculator does div
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• Install R Commander from the R
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- In large random samples (n 1 and
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Course Notes - Department of Mathematics and Statistics

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