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INFERENCE • Inference – extension of results obtained from an experiment (sample) to the general population • use of sample data to draw conclusions about entire population • Parameter – number that describes a population – Value is not usually known – We are unable to examine population • Statistic – number computed from sample data – Estimate unknown parameters – Computed to estimate unknown parameters • Mean, standard deviation, variability, etc.. Notations population mean sample mean INFERENCE • Sampling variability – natural variation of outcomes from an experiment that results from inherent differences among samples – no two samples are going to give same statistics • E.g. Assume that you have a population of 1000 point locations across a study area and you wish to estimate the mean value of soil erosion across the study area using this sample of 10. Choose a simple random sample this population. How many different samples can be chosen? 23 1000C10= 2.6 x 10 INFERENCE • How can experimental results be trusted? If x is rarely exactly right and varies from sample to sample, why is it nonetheless a reasonable estimate of the population mean μ? • How can we describe the behavior of the statistics from different samples? – E.g. the mean value ESTIMATION OF MEAN Example: Sulfur compounds such as dimethyl sulfide (DMS) are sometimes present in wine. DMS causes “off-odors” in wine, so winemakers want to know the odor threshold, the lowest concentration of DMS that the human nose can detect. Different people have different thresholds, so we start by asking about the mean threshold in the population of all adults. To estimate the population mean, we present tasters with both natural wine and the same wine spiked with DMS at different concentrations to find the lowest concentration at which they identify the spiked wine. • How can we estimate the mean threshold value for the population? LAW OF LARGE NUMBERS 1) If we keep taking larger and larger samples, the statistic is guaranteed to get closer and closer to the parameter value. SAMPLING DISTRIBUTIONS 2) How else can we estimate the population mean value, if we can not take very large samples for our study? e.g. What can we say about the estimate of mean from say 10 subjects as an estimate of μ? Here are the odor thresholds (micrograms of DMS per liter of wine) for 10 randomly chosen subjects: 28 • 40 28 33 20 31 29 27 17 21 How well would our mean value from this sample estimate the true parameter value?