DOC PREVIEW
UVA STAT 2120 - Topic_06

This preview shows page 1-2-23-24 out of 24 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Introduction to InferenceIntroduction to InferenceEstimating with ConfidenceSection 6.1General approach to inferenceProbability calculations help distinguish patterns seen in data between those that are due to chance and those that reflect a real feature of the phenomenon under studyExample:Weights of brown eggs is N(65, 5) gramsExample:Weights of brown eggs is N(65, 5) gramsSelect (by SRS) a dozen white eggs. SupposeHow do white eggs compare to brown eggs?Treat as known(more later)Note: The setup has population distribution N(μ, σ= 5), hence is N(μ, )Target of inferenceσ/√n = 5/√12Target of inference(mean white egg weight)Types of inference A confidence interval (CI) supplements an estimate of a parameter with an indication of its variabilitypy A significance test assesses the truth of a hypothesis about a parameter by comparing it with observed dataBoth work byreporting probabilities that describe whatBoth work by reporting probabilities that describe what would happen “in the long run,” if the experiment was repeated many, many timesConfidence intervals By the central limit theorem, it is natural to state distances of from μin units ofμ A confidence level for within units from μ isSt tC100% “ fid ” th tli ithi th b dState C100% “confidence” thatμlies within the bounds“Margin of error”(smaller is better)InterpretationIn the “long run,” C100% of confidence intervalswould cover μExample: White egg weightsThe mean of a sample of n = 12 white eggs is N(μ) gramsN(μ, ) grams.Observea = 3/1.44 = 2.08Consider a margin of error of grams State 96.23% confidence that μ lies within the boundsConfidence interval formulasGiven a desired confidence level, C, a corresponding C100% approximate* CI for μisppμ“Critical value” The quantity z* is such that C = P(-z* ≤ Z ≤ z* ), where Z is N(0,1).  The associated margin of error is m = z* σ / √n* The confidence level is exact if the population is Normal, and approximately correct if n is largeExample: White egg weights (continued)The mean of a sample of n = 12 white eggs is N(μ) gramsN(μ, ) grams.ObserveFor 95% confidence, deduce P(Z ≤ -1.96) = 0.025 = (1 – C)/2, ⇒ P(-1.96 ≤Z≤1.96) = 0.95 = C⇒ z* = 1.96.St t 95% fid th tli ithi th b dState 95% confidence thatμlies within the boundsElementary behavior of confidence intervalsFocus on margin of error m = z* σ / √n Requiring higher confidence increases m Example: 95% CI is (61.37, 67.03)96.23% CI is (61.2, 67.2)Smaller population standard deviationσdecreasesmSmaller population standard deviation, σ, decreases mCannot control Larger sample size, n, decreases mSet in advanceSet in advanceExample: Loan-to-deposit ratioThe st. dev. of the loan-to-deposit (LTDR) ratio among a certain population of banks is known to be σ= 12.3.pp In a sample of n = 110 such banks, the mean is ⇒acorresponding 95% CI has boundsm = 2.3⇒acorresponding 95% CI has bounds3 In a separate sample of n = 25 such banks, the mean ism=48⇒ a corresponding 95% CI has boundsm= 4.8Selecting a sample sizeA sample size may be chosen to target a desired margin of error Desired confidence level, C, provides z* Use only when planning a sample, not after the factExample: Loan-to-deposit ratio (continued)In planning a new sample of banks, what sample size is needed for a margin of error no more than m = 3, with g,95% confidence?Recall σ = 12.3.Round up ton=65 banksRound up to n 65 banksCautions about confidence intervals The sample must have been drawn by SRS Since the mean is not resistant, neither is the CIAbsent e treme o tliers or strongske nessn≥15isAbsent extreme outliers or strong skewness, n≥15 is sufficient for accuracy For σ unknown, replace with s. The CI will be accurate if n is large. (More later) The margin of error does not accommodate errors due to sampling biasIntroduction to InferenceIntroduction to InferenceTests of SignificancegSection 6.2Basic approach to significance testing State a comparison of hypotheses, H0versus Ha Null hypothesis: H0, status quoyp0,q Alternative hypothesis: Ha, suspicion Select a test statistic and calculate a P-value Measures the compatibility of the data with H0 Compare the P-value to desired significance levelHypothesesExample: Weights of brown eggs is N(65, 5) gramsHow do white eggs compare to brown eggs?Possibilities: Suspect that white eggs weigh less, on average H0: μ≥65 versusHa: μ< 650μ65esusaμ65 Suspect that white eggs weigh more, on averageH0:μ≤65versusH:μ>65H0: μ≤65 versusHa: μ> 65 Suspect different weights, on averageH:μ=65versusH:μ≠65H0: μ= 65 versusHa: μ≠65Guidelines for formulating H0versus Ha Each of H0and Hais expressed in terms of a parameter (e.g., μ)p(g,μ) Hait is typically what you hope to establish is true H0versus Hamay be:One-sided: H0: μ≥μ0versus Ha: μ < μ0“=” is always in H0H0: μ≤μ0versus Ha: μ > μ0Two-sided: H0: μ = μ0versus Ha: μ≠μ0 Focus on Haand set H0to what is not HaTest statistic The test statistic is typically an unbiased estimate of the parameter relevant to H0versus Hap0a for hypotheses about μ The observed distance of the test statistic from H0indicates evidence against H0 The direction away from H0is determined by Ha.y0yaExample: How do white eggs compare to brown eggs?H0: μ≥ 65 versus Ha: μ < 65Evidence against H0when is smallHa: Do white eggs weigh less?P-values A P-value measures how surprising the patterns in the data would be if H0was true0 A smaller P-value indicates a more surprising patternCalculate as the probability of observing data“at leastCalculate as the probability of observing data at least as extreme” as was observed if H0was trueExample:How do white eggs compare to brown eggs?Example:How do white eggs compare to brown eggs?H0: μ≥65 versus Ha: μ < 65F12 ith5bFrom n = 12 eggs, with σ= 5 grams, observeSignificance level The significance level is the decisive level, α, at which the P-value is small enough to “reject H0”gj0 The data are statistically significant when the P-value equals or falls belowαvalue equals or falls below α Typical value is α≈0.05Example:How do white eggs compare to brown eggs?Example:How do white eggs compare to brown eggs?H0: μ≥65


View Full Document

UVA STAT 2120 - Topic_06

Download Topic_06
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Topic_06 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Topic_06 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?