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3.3 Toward Statistical Inference

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What is statistical inference? Statistical inference is using a fact about a sample to estimate the truth about the whole population.

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Parameter vs. Statistic A numerical property of a population is called a parameter. A numerical property of a sample is called a statistic. A numerical property of a population is called a parameter. A numerical property of a sample is called a statistic. E.g. The average SAT score for all high school students who took the SAT is a parameter. The average SAT score of 10,000 particular high school students is a statistic. A proportion of a population is denoted p. A proportion of a sample is denoted p.

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Sampling Variability The value of the statistic p will change from sample to sample. This is the idea of sampling variability. The value of the statistic p will change from sample to sample. This is the idea of sampling variability. Hopefully if we take lots of random samples of the same size from the same population, the variation from sample to sample will follow a predictable pattern.

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Sampling Distribution The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. This is an interesting concept. If our population is comprised of people, then the sampling distribution of sample size 100 is the collection of ALL 100 combinations of people. The idea is that the sampling distribution statistics should match the population statistics (we’ll see more about when and how in a latter chapter). But repeated sampling of the same sample size should give us a good idea about the population parameters.

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Suppose we are studying gestational length in pregnancies in Montgomery County, and we take 110 random samples of 1000 women who have given birth.

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The idea here is that it is beginning to look approximately normal so that there is good reason to believe that average length of pregnancy in Montgomery County is around 264 days.

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Bias A statistic is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. Of course, under this definition most statistics would be biased, and even if a statistic is unbiased, there is no way to know it! The idea is to reduce bias.

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Variability The variability of a statistic is described by the spread of its sampling distribution. The larger the probability sample, the smaller the spread. Thus, we strive for low bias and low variability. This is realized through repeated sampling and large sample size.

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