Lecture 46.1. Uncertainty and confidenceSlide 3Slide 4The weight of single eggs of the brown variety is normally distributed N(65 g,5 g). Think of a carton of 12 brown eggs as an SRS of size 12.Confidence intervalImplicationsRewordedSlide 9Varying confidence levelsHow do we find specific z* values?Link between confidence level and margin of errorDifferent confidence intervals for the same set of measurementsImpact of sample sizeSample size and experimental designWhat sample size for a given margin of error?Cautions:Section 6.2: Tests of SignificanceSlide 19Null and alternative hypothesesSlide 21One-sided and two-sided testsHow to choose?The P-valueInterpreting a P-valueSlide 26Tests for a population meanSlide 28Slide 29The significance level aSlide 31Rejection region for a two-tail test of µ with α = 0.05 (5%)Confidence intervals to test hypothesesLogic of confidence interval testSection 6.3: Use and abuse of testsCaution about significance testsPractical significanceSlide 38Interpreting effect size: It’s all about contextThe power of a testSlide 41Slide 42Factors affecting power: Size of the effectSlide 44Slide 45Slide 46Type I and II errorsSlide 48Section 7.1 Inference for the mean of a populationSlide 50When s is unknownStandard deviation s – standard error s/√nThe t distribution:Slide 54Standardizing the data before using Table DTable DTable A vs. Table DThe one-sample t-confidence intervalSlide 59Slide 60The one-sample t-testSlide 62Table D How to:Slide 64Slide 65Slide 66Matched pairs t proceduresSlide 68Slide 69Does lack of caffeine increase depression?Slide 71Slide 72RobustnessPower of the t-testSlide 75Slide 76Inference for non-normal distributionsTransforming dataNonparametric method: the sign testSection 8.1Sampling distribution of p^ — reminderConditions for inference on pLarge-sample confidence interval for pMedication side effectsSlide 85Slide 86Slide 87Interpretation: magnitude vs. reliability of effectsSample size for a desired margin of errorSlide 90Lecture 4 Statistical Inference. Inference for one population mean and one population proportion6.1. Uncertainty and confidenceAlthough the sample mean, , is a unique number for any particular sample, if you pick a different sample you will probably get a different sample mean. In fact, you could get many different values for the sample mean, and virtually none of them would actually equal the true population mean, .xBut the sample distribution is narrower than the population distribution, by a factor of √n.Thus, the estimates gained from our samples are always relatively close to the population parameter µ.nSample means,n subjectsnPopulation, xindividual subjectsx x If the population is normally distributed N(µ,σ), so will the sampling distribution N(µ,σ/√n),Red dot: mean valueof individual sample95% of all sample means will be within roughly 2 standard deviations (2*/√n) of the population parameter Because distances are symmetrical, this implies that the population parameter  must be within roughly 2 standard deviations from the sample average , in 95% of all samples.This reasoning is the essence of statistical inference.nxThe weight of single eggs of the brown variety is normally distributed N(65 g,5 g).Think of a carton of 12 brown eggs as an SRS of size 12..You buy a carton of 12 white eggs instead. The box weighs 770 g. The average egg weight from that SRS is thus = 64.2 g.  Knowing that the standard deviation of egg weight is 5 g, what can you infer about the mean µ of the white egg population? There is a 95% chance that the population mean µ is roughly within ± 2/√n of , or 64.2 g ± 2.88 g.population sample What is the distribution of the sample means ? Normal (mean , standard deviation /√n) = N(65 g,1.44 g). Find the middle 95% of the sample means distribution.Roughly ± 2 standard deviations from the mean, or 65g ± 2.88g. x x xConfidence intervalThe confidence interval is a range of values with an associated probability or confidence level C. The probability quantifies the chance that the interval contains the true population parameter. ± 4.2 is a 95% confidence interval for the population parameter . This equation says that in 95% of the cases, the actual value of  will be within 4.2 units of the value of . x xImplicationsWe don’t need to take a lot of random samples to “rebuild” the sampling distribution and find  at its center. nnSamplePopulationAll we need is one SRS of size n and relying on the properties of the sample means distribution to infer the population mean .RewordedWith 95% confidence, we can say that µ should be within roughly 2 standard deviations (2*/√n) from our sample mean bar.In 95% of all possible samples of this size n, µ will indeed fall in our confidence interval.In only 5% of samples would be farther from µ. nx xA confidence interval can be expressed as:Mean ± m m is called the margin of error within ± mExample: 120 ± 6Two endpoints of an interval  within ( − m) to ( + m) ex. 114 to 126 A confidence level C (in %) indicates the probability that the µ falls within the interval. It represents the area under the normal curve within ± m of the center of the curve.mmx x xConfidence intervals contain the population mean  in C% of samples. Different areas under the curve give different confidence levels C. Example: For an 80% confidence level C, 80% of the normal curve’s area is contained in the interval.Cz*−z*Varying confidence levelsPractical use of z: z* z* is related to the chosen confidence level C. C is the area under the standard normal curve between −z* and z*.x z *nThe confidence interval is thus:How do we find specific z* values?We can use a table of z/t values (Table C). For a particular confidence level, C, the appropriate z* value is just above it. We can use software. In R: qnorm(probability,mean,standard_dev) gives z quantile for a given probability.Since we want the middle C probability, the probability we need to input is (1 - C)/2 Example: For a 98% confidence level, qnorm(0.01,0,1) = −2.326348 (= neg. z*)Example: For a 98% confidence level, z*=2.326Link between confidence level and margin of errorThe confidence level C determines the value of z* (in table C).The margin of