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Reminders: Parameter – number that describes the population Statistic – number that is computed from the sample data Mean of population = µ Mean of sample = x

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4.3 Sample Distributions Statistical inference uses sample data to draw conclusions about the entire population

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Statistical estimation and the law of large numbers As the number of observations drawn increases, the mean x of the observed values gets closer and closer to the mean of the population Similar to flipping a coin “many times” Foundation of gambling casinos and insurance companies

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Sampling distributions the sampling distribution of a statistic is the distribution (histogram) of values taken by the statistic in all possible samples of the same size from the same population

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Hospital beds data Make a histogram of the data & calculate the population mean Take 2 samples of ten and find the sample means Make a histogram of all the sample means Compare the population mean to the approximate mean on the histogram Take two more samples of ten and find the sample means. Make histogram of all sample means and compare to population mean

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Central limit theorem As the sample size increases, the distribution of x changes shape: it looks less like that of the population and more like a normal distribution (no matter what the population curve looks like) When the sample is large enough, the distribution of x is very close to normal

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Hospital beds data Take 2 samples of 20 and find the sample means Make a histogram of all the sample means Compare the population mean to the approximate mean on the histogram Compare the histogram made with 2 samples of twenty to the histogram made with 2 samples of ten

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When we choose many SRS’ s from a population, the sampling distribution of the sample means... is centered at the mean of the original population is less spread out than the distribution of individual observations The mean of the sampling distribution of x is μ and its standard deviation is

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Unbiased estimator – “correct on the average in many samples” because the mean of x is equal to μ, we say that the statistic x is an unbiased estimator of the parameter μ The standard deviation of the distribution of x gets smaller as we take larger samples

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Engineers are redesigning fighter jet ejection seats to better accommodate women. In human engineering and product design, it is often important to consider people’s weights so that airplanes or elevators aren’t overloaded, chairs don’t break, etc. Given that the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, find the probability that If 1 woman is randomly selected, her weight is greater than 150 lb If 36 different women are randomly selected, their mean weight is greater than 150 lb

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Interpretation There is a 0.405 probability that a woman will weigh more than 150 lb, but there is only a 0.0738 probability that 36 women will have a mean weight of more than 150 lb. It is much easier for an individual to deviate from the mean than it is for a group of 36. A single extreme weight among the 36 weights will have reduced impact when it is averaged in with the other weights.

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IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. If 25 people are randomly selected for an IQ test, find the probability that their mean IQ score is between 95 and 105.

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