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Regardless of the shape of the population (random variable X), the shape of the distribution of the sample means (the sampling distribution of “x-bar”) will be approximately normal if the sample size n ≥ 30. How large does the sample size, “n, have to be before the distribution of the sample means is approximately normal? What does the “Central Limit Theorem” tell us?Īccording to the “Central Limit Theorem”, regardless of the shape of the population (random variable X), the shape of the distribution of the sample means (the sampling distribution of “x-bar”) becomes approximately normal as the sample size n increases. If a random variable X is normally distributed, the distribution of the sample means, “x-bars”, is automatically normally distributed. If a random variable X is normally distributed, what do we know about the distribution of the sample means, “x-bars” The standard deviation of the sampling distribution of “x-bar” is called the “standard error of the mean”. What is the standard deviation of the sampling distribution of “x-bar” called? The “sampling distribution of x-bar” will have mean, µ(x-bar) = µ, and standard deviation, σ(x-bar) = σ/sqrt(n). Given a simple random sample of size n drawn from a large population with mean  and standard deviation , what do we know about the “mean” and “standard deviation” of the sampling distribution of x-bar? The sampling distribution of the sample mean, “x-bar”, is the probability distribution of ALL possible values of the random variable, x-bar, computed from a sample of size “n” taken from a population with mean  and standard deviation . What do we mean by the “sampling distribution of the sample mean (x-bar)”? The sampling distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n. What do we mean by the “sampling distribution” of a statistic? That is, the sample means have a “shape”, “center” and “spread”. Just like any other random variables, the “x-bars” have probability distributions associated with them. Yes, x-bar is a random variables because its value varies from sample to sample.ĭo the sample means, “x-bars”, have an associated probability distribution? Is a sample mean, “x-bar”, a random variables? Why? Concepts of Sampling distributions of means and proportions Question