Course Notes - Department of Mathematics and Statistics

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95% confidence and prediction intervals for tmp.lmx42●y0●●●●●●●●●●●●●●observedfitconf intpred int●●−2●−4x−4 −2 0 2 4x• For our height example s e = √ 103.39 = 10.168• If make the prediction for a father of 175cm tall then :√sŷ = s e 1 + 1 n + (x k − ¯x) 2Σ(x i − ¯x) 2√sŷ = 10.168 1 + 1 (175 − 178.493)2+309 23485.35sŷ = 10.1871• We use the same t-value that was used in the confidence of slope.(t 307 = 1.96)• So the prediction interval isŷ 175 ± t 307 × sŷ172.046 ± 1.96(10.1871)172.046 ± 19.9667152.0793 < ŷ 175 < 192.0127• Important Note: While the process for constucting a confidenceinterval and a prediction interval is identical, there is a conceptualdifference.222
• A confidence interval estimates an unknown population parameter.• A prediction interval instead estimates the potential data valuefor an individual.10.4 CorrelationA note on extrapolation• It is risky to use a regression equation to predict values outsidethe range of the observed data.• This process is known as extrapolation• The reason that it is risky is that there is no guarantee that therelationship will continue beyond the range of observed data.100 metre example100m World Record Progression●●Time(seconds)9.6 9.7 9.8 9.9 10.0●●● ●●●●● ●●●● ●●●●●1970 1980 1990 2000 2010Year223
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Contents1 Course Administration 42
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10 Regression 20210.1 Introduction
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Resource PageDepartment of Mathemat
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3 Data and Study Designs3.1 Basic d
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Parameter• Parameter: Fixed numbe
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TypesDiscrete - can put in one-to-o
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- How do geologists know?- Sediment
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Source: skyscrapercity.comSampling
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- 1 in 3.8 million - 3.8 million di
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Important designs• Completely ran
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B. Randomised control - of all chil
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- Reason why observational studies
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Things that go wrong• Even the be
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4 ProbabilityFred’s DayFred awoke
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• The event that the mouse we tra
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Blood donor example - Multiplicatio
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Fair Die Example• A fair die is t
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Tree diagram rules• Add Verticall
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Tree Diagrams - Independent Stages0
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Dependent Stages• Andrew, John an
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0.90BBiopsy +ve (true positive)0.00
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Calculating Probabilities• Estima
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4.3 Random VariablesRandom Variable
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Calculating the Variance• The sam
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Examples• Consider a data set XX
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Combining 2 Random Variables• If
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Fred’s DayShould Fred be worried
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Now using the complementary event r
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Mean of Binary DistributionThe mean
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• Mean number of successes• Var
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Using the Formulan = 3, x = 2, π =
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• A probability less than 0.05 is
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Endangered bird egg example solutio
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RELATIVE FREQUENCY HISTOGRAMNormal
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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
Page 172 and 173: RESIDUAL MEAN SQUARE (s 2 e)Group A
Page 174 and 175: 9.2 Post ANOVA AnalysisPOST ANOVA A
Page 176 and 177: • Hence no need to additionally c
Page 178 and 179: Since zero is excluded, and the int
Page 180 and 181: SUM OF SQUARESTotal SS = 93 2 + 100
Page 182 and 183: General Level SS = 12 x ( 9812 )2=
Page 184 and 185: ANOVA TABLEA one factor analysis of
Page 186 and 187: Data > Manage variables in active d
Page 188 and 189: DATA COMPONENTSEach data value can
Page 190 and 191: ANOVA TABLE - Two Factor FactorialS
Page 192 and 193: A B C D EChristchurch 120.5 119 122
Page 194 and 195: DataFERTILISER LEVELSEED TYPE Low M
Page 196 and 197: Interaction PlotsHypothesis test fo
Page 198 and 199: Fred’s WeightUsing the informatio
Page 200 and 201: Fred wanted to address the followin
Page 202 and 203: 10 RegressionFred’s Dog/BeardFred
Page 204 and 205: 2. studies which have measured bina
Page 206 and 207: • What if we colour code the poin
Page 208 and 209: Statistically• We use slightly di
Page 210 and 211: Blood pressure exampleStress(x) Blo
Page 212 and 213: Focussed look on the limits of our
Page 214 and 215: Height example• We will perform a
Page 216 and 217: Normality Assumption FAILPP PlotExp
Page 218 and 219: 10.3 Confidence Intervals and Regre
Page 220 and 221: 0.236 < β 1 < 0.496• We can conc
Page 224 and 225: 100m World Record Progression●●
Page 226 and 227: Negative correlation• The denomin
Page 228 and 229: Stress exampleStress(x) Blood Press
Page 230 and 231: 10.5 Multiple regression• Simple
Page 232 and 233: • Appears that total lung capacit
Page 234 and 235: Multiple linear regression predicti
Page 236 and 237: Three variable modelIntepreting Mod
Page 238 and 239: • So for men we have:y = −7.066
Page 240 and 241: • Is the variable age an importan
Page 242 and 243: Extra sum of squares principle• T
Page 244 and 245: Diagnostics• Remember our Assumpt
Page 246 and 247: TableTreatmentControlBP(Y) AGE(X) B
Page 248 and 249: • So for people in the control gr
Page 250 and 251: Discussion• The value of d is the
Page 252 and 253: Multiple Linear RegressionTreatment
Page 254 and 255: • Logistic regression is sometime
Page 256 and 257: • Given the s.e.(OR) = 0.751. Thi
Page 258 and 259: • We get the same as when we calc
Page 260 and 261: Female(0)Male(1)Born Left(1) Right(
Page 262 and 263: Fred’s Dog/BeardTo help solve his
Page 264 and 265: This is a very high r 2 value and i
Page 266 and 267: Attractiveness = 4.85 − 0.03 × (
Page 268 and 269: • Because the calculator does div
Page 270 and 271: • 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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