Course Notes - Department of Mathematics and Statistics

More documents

Recommendations

Info

X = x i Pr(X = x i )0 0.101 0.252 0.403 0.204 0.05Meanµ X = 0(0.10) + 1(0.25) + 2(0.40) + 3(0.20) + 4(0.05) = 1.85Varianceσ 2 X = (0 − 1.85)2 × 0.10 + (1 − 1.85) 2 × 0.25 + (2 − 1.85) 2 ×0.40 + (3 − 1.85) 2 × 0.20 + (4 − 1.85) 2 × 0.05 = 1.0275Standard Deviationσ X =√σ 2 X = √ 1.0275 = 1.0137Combining Random Variables• Often we are interested in the mean and the variance of a rescaledrandom variable, or in the mean and variance of sums (or differences)of random variables.• We will look at some properties of random variables (both discreteand continuous)Modifying Random Variables• If X is an independent random variable and a and b are constants.• Consider a new random variable Y, where:Y = a + bX• The mean of Y can be calculated by:µ Y = a + bµ X• The variance of Y can be calculated by:σY 2 = b2 σX252
Examples• Consider a data set XX = 3, 4, 5, 6, 7• So µ X = 5 and σX 2 = 2.5• Now consider Y = 2 × XX = 6, 8, 10, 12, 14• In this case a = 0 and b = 2 soµ Y = a + bµ X = 0 + 2 × 5 = 10σY 2 = b2 σX 2 = 22 × 2.5 = 10• So µ X = 5 and σX 2 = 2.5• Now consider Y = 4 × X + 3X = 15, 19, 23, 27, 31• In this case a = 3 and b = 4 soµ Y = a + bµ X = 3 + 4 × 5 = 23σY 2 = b2 σX 2 = 42 × 2.5 = 40Temperature Problem• Temperatures can be recorded in degrees Fahrenheit. Suppose arandom variable F measures January temperature ( ◦ F) in Dunedin.• Daily maximum summer temperatures have a mean of 70 ◦ F witha standard deviation of 5 ◦ F.• Use the conversion formula C = 5 9(F-32) to find the mean andstandard deviation for the temperature in degrees Celsius.53
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 and 18: Source: skyscrapercity.comSampling
Page 19: - 1 in 3.8 million - 3.8 million di
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: Calculating the Variance• The sam
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
Page 75 and 76: • Find P r(−1 < Z < 1.64)pnorm(
Page 77 and 78: INVERSE PROBLEMS USING R-COMMANDER
Page 79 and 80: CALCULATING PROBABILITIESAssume tha
Page 81 and 82: CONTINUITY CORRECTION• Normal pro
Page 83 and 84: Sample size = 10Consider the situat
Page 85 and 86: ConclusionWhat conclusion would you
Page 87 and 88: 10%5.5 6.0 6.5 7.0 7.5 8.0 8.5XWe k
Page 89 and 90: 6 Sampling Distributions and Estima
Page 91 and 92: DERIVATION IWe can think of the sam
Page 93 and 94: Solution IIPr(X > 172) = 1 - pnorm(
Page 95 and 96: qnorm(0.975) as 2.5% of the area is
Page 97 and 98: 6.2.1 Sample Size CalculationEXAMPL
Page 99 and 100: The 99% Confidence IntervalThe 99%
Page 101 and 102: 6.3 Comparing Two SamplesCOMPARING
Page 103 and 104:
THE POOLED VARIANCEIf the variances
Page 105 and 106:
SOLUTION - Calculating the confiden
Page 107 and 108:
Confidence Interval for raw dataTo
Page 109 and 110:
= 6.6 ± 24.6That is, -18.0 < µ in
Page 111 and 112:
STANDARD DEVIATIONIf P = X n : σP
Page 113 and 114:
6.4.2 Sample Size CalculationEXAMPL
Page 115 and 116:
Fred for MayorFred has decided to t
Page 118 and 119:
Fred’s mate Bob pointed out that
Page 120 and 121:
ALTERNATIVE HYPOTHESESTwo types:Stu
Page 122 and 123:
• If we use the sample standard d
Page 124 and 125:
Calculate the test statistic:Test s
Page 126 and 127:
Finding the pooled variance:Recall
Page 128 and 129:
Caluclate the estimated standard er
Page 130 and 131:
7.6 Significance and Conclusiveness
Page 132 and 133:
Conclusion: There is no evidence th
Page 134 and 135:
Some practical pointers• Aim for
Page 136 and 137:
Fred’s Parking & Extracurricular
Page 138 and 139:
Fred calculated the p-value using t
Page 140 and 141:
Fred told Carol he would use odds r
Page 142 and 143:
Flu vaccine example• There were 1
Page 144 and 145:
8.3 Attributable Risk (AR)The attri
Page 146 and 147:
• Therefore w x ≈ww+x and y z
Page 148 and 149:
Aspirin study example - gastrointes
Page 150 and 151:
Mobile phone example• Human expos
Page 152 and 153:
OR = 0.81 with CI (0.39,1.70)• p
Page 154 and 155:
• We calculate these expected cou
Page 156 and 157:
• We can calculate the p value us
Page 158 and 159:
χ 2 =(100 − 108.99)2 (107 − 11
Page 160 and 161:
• The reason for the discrepancy
Page 162 and 163:
Summary - Confidence IntervalsFacto
Page 164 and 165:
Since 1 is excluded from this 95% i
Page 166 and 167:
9 ANOVAFred’s Public ImageSince F
Page 168 and 169:
• Previously used two sample t-te
Page 170 and 171:
• 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 222 and 223:
95% confidence and prediction inter
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
Page 272 and 273:
- In large random samples (n 1 and
show all

Course Notes - Department of Mathematics and Statistics

Create successful ePaper yourself

Delete template?

Save as template?