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A.P. Statistics Chapter 8 Key facts [8-1] Definitions Any quantity computed from values in a sample is called a statistic. The observed value of a statistic depends on the particular sample selected from the population; typically, it varies from sample to sample. This variability is called sampling variability.The distribution of the possible values of a statistic is called its sampling distribution. [8-2] General Properties of the Sampling Distribution of x Let x denote the mean of the observations in a random sample of size n from a population having mean μ and standard deviation σ. Denote the mean value of the x distribution by x and the standard deviation of the x distribution by x . Then the following rules hold. Rule 1. x x Rule 2. This rule is exact if the population is infinite, and is approximately n correct if the population is finite and no more than 10% of the population is included in the sample. Rule 3. When the population distribution is normal, the sampling distribution of x is also normal for any sample size n. Rule 4. (Central Limit Theorem) When n is sufficiently large, the sampling distribution of x is well approximated by a normal curve, even when the population distribution is not itself normal. *If n is large or the population distribution is normal, the standardized variable z x x x x n has (at least approximately) a standard normal (z) distribution. *The Central Limit Theorem can safely be applied if n exceeds 30. Illustration of the CLT(Central Limit Theorem) NORMAL POPULATION Normal distribution of platelet size x with = 8.25 and = .75 Sample histograms for x based on 500 samples, each consisting of n observations: (a) n = 5; (b) n = 10; (c) n = 20; (d) n = 30 NON-NORMAL POPULATION The population distribution (μ = 9.841) Four histograms of 500 x values for Example 8.5 [8-3] General Properties of the Sampling Distribution of p Let p be the proportion of successes in a random sample of size n from a population whose proportion of S’s is π. Denote the mean value of p by p and the standard deviation by p . Then the following rules hold. Rule 1. p Rule 2. p (1 ) This rule is exact if the population is infinite, and is n approximately correct if the population is finite and no more than 10% of the population is included in the sample. Rule 3. When n is large and π is not too near 0 or 1, the sampling distribution of p is approximately normal. *The farther the value of π is from .5, the larger n must be for a normal approximation to the sampling distribution of p to be accurate. A conservative rule of thumb is that if both nπ 12 Histograms of 500 values of p ( = .07) Illustration of the sampling distribution of p Histograms of 500 values of p ( = .07/population positively skewed)