A review of key statistical conceptsAn overview of the reviewPopulations and ParametersSlide 4ParametersSamples and StatisticsStatisticsExample: Smoking at PSU?Example: Grade inflation?Example: A linear relationship?Two ways to learn about a population parameterConfidence intervalsThe situationConfidence intervals for proportions in newspapersGeneral form of most confidence intervals(1-α)100% t-interval for population mean Determining the t-multiplierTypical confidence coefficientst-interval for mean in MinitabLength of confidence intervalHow length of CI is affected?Hypothesis testingGeneral idea of hypothesis testingExample: Normal body temperatureMaking the decisionMaking the decision (cont’d)Again, idea of hypothesis testing: criminal trial analogyCriminal trial analogy (continued)Slide 29Slide 30Very important pointErrors in criminal trialsErrors in hypothesis testingDefinitions: Types of errorsSlide 35Possible hypotheses about mean µCritical value approachRight-tailed critical valueLeft-tailed critical valueTwo-tailed critical valueP-value approachRight-tailed P-valueLeft-tailed P-valueTwo-tailed P-valueExample: Right-tailed testExample: Right-tailed critical valueExample: Right-tailed P-valueExample: Left-tailed testExample: Left-tailed critical valueExample: Left-tailed P-valueExample: Two-tailed testExample: Two-tailed critical valueExample: Two-tailed P-valueA review of key statistical conceptsAn overview of the review•Populations and parameters•Samples and statistics•Confidence intervals•Hypothesis testingPopulations and Parameters… and Samples and StatisticsPopulations and Parameters•A population is any large collection of objects or individuals, such as Americans, students, or trees about which information is desired.•A parameter is any summary number, like an average or percentage, that describes the entire population.Parameters•Examples:–population mean µ = average temperature–population proportion p = proportion approving of president’s job performance•99.999999999999….% of the time, we don’t (...or can’t) know the real value of a population parameter.•Best we can do is estimate the parameter!Samples and Statistics•A sample is a representative group drawn from the population.•A statistic is any summary number, like an average or percentage, that describes the sample.Statistics•Examples–sample mean (“x-bar”)–sample proportion (“p-hat”)•Because samples are manageable in size, we can determine the value of statistics.•We use the known statistic to learn about the unknown parameter.Example: Smoking at PSU?Population of 42,000 PSU studentsWhat proportion smoke regularly?Sample of 987 PSU students43% reported smoking regularlyExample: Grade inflation?Population of 5 million college studentsIs the average GPA 2.7?Sample of 100 college studentsHow likely is it that 100 students would have an average GPA as large as 2.9 if the population average was 2.7?Example: A linear relationship?424140393837363534350030002500Gestation (weeks)Birth weight (grams)S = 167.327 R-Sq = 77.5 % R-Sq(adj) = 76.8 %Weight = -2037.00 + 130.817 GestationRegression PlotY-hat = a + b XE(Y) = A + B XPopulation lineSample estimateTwo ways to learn about a population parameter•Confidence intervals estimate parameters.–We can be 95% confident that the proportion of Penn State students who have a tattoo is between 5.1% and 15.3%.•Hypothesis tests test the value of parameters.–There is enough statistical evidence to conclude that the mean normal body temperature of adults is lower than 98.6 degrees F.Confidence intervalsA review of conceptsThe situation•Want to estimate the actual population mean .•But can only get “x-bar,” the sample mean.•Use “x-bar” to find a range of values, L<<U, that we can be really confident contains . •The range of values is called a “confidence interval.”Confidence intervals for proportions in newspapers•“Sample estimate”: 69% of 1,027 U.S. adults think using a hand-held cell phone while driving a car should be illegal.•The “margin of error” is 3%.•The “confidence interval” is 69% ± 3%.•We can be really confident that between 66% and 72% of all U.S. adults think using a hand-held cell phone while driving a car should be illegal.Source: ABC News Poll, May 16-20, 2001General form of most confidence intervals•Sample estimate ± margin of error•Lower limit L = estimate - margin of error•Upper limit U = estimate + margin of error•Then, we’re confident that the value of the population parameter is somewhere between L and U.(1-α)100% t-interval for population mean nsx1,21 ntFormula in notation:Formula in words:Sample mean ± (t-multiplier × standard error)Determining the t-multiplier43210-1-2-3-40.40.30.20.10.0t(14)density122Typical confidence coefficientsConf. coefficient Conf. level0.90 90% 0.950.95 95% 0.9750.99 99% 0.99521 %1001 1t-interval for mean in MinitabOne-Sample T: FVCVariable N Mean StDev SE Mean 95.0% CI FVC 8 3.5875 0.1458 0.0515 (3.4655,3.7095)We can be 95% confident that the mean forced vital capacity of all female college students is between 3.5 and 3.7 liters.Length of confidence interval•Want confidence interval to be as narrow as possible.•Length = Upper Limit - Lower LimitHow length of CI is affected?•As sample mean increases…•As the standard deviation decreases…•As we decrease the confidence level…•As we increase sample size …nsx tHypothesis testingA review of conceptsGeneral idea of hypothesis testing•Make an initial assumption.•Collect evidence (data).•Based on the available evidence (data), decide whether to reject or not reject the initial assumption.Example: Normal body temperaturePopulation of many, many adultsIs average adult body temperature 98.6 degrees? Or is it lower?Sample of 130 adultsAverage body temperature of 130 sampled adults is 98.25 degrees.Making the decision•It is either likely or unlikely that we would collect the evidence we did given the initial assumption.•If it is likely, then we “do not reject” our initial assumption. There is not enough evidence to do otherwise.Making the decision (cont’d)•If it is unlikely, then:–either our initial assumption is correct and we experienced a
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