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Stats

Stats. Section 5.5 Notes. Remember the following:. A random variable is a variable that has a single numerical value for each outcome of a procedure. A probability distribution is a graph, table, or formula that gives the probability for each value of a random variable.

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Stats

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  1. Stats Section 5.5 Notes

  2. Remember the following: • A random variable is a variable that has a single numerical value for each outcome of a procedure. • A probability distribution is a graph, table, or formula that gives the probability for each value of a random variable. • The sampling distribution of the mean is the probability distribution of the sample means, will all samples having the same size n. • It was determined in Section 5.4 that the mean of the sample means is equal to the mean of the population. As the sample size increases, the sample means tend to vary less.

  3. The Central Limit Theorem • The central limit theorem states that if the sample size is large enough, the distribution of the sample means can be approximated by a normal distribution, even if the original population is not normally distributed.

  4. The Central Limit Theorem and the Sampling Distribution of Given: • The random variable x has a distribution, which may or may not be normal, with mean and standard deviation • Simple random samples all of the same size n are selected from the population. The samples are selected so that all possible values of size n have the same chance of being selected.

  5. Conclusions: • The distribution of sample means will, as the sample size increases, approach a normal distribution. • The mean of all sample means is the population mean • The standard deviation of all sample means is

  6. Practical Rules Commonly Used • If the original population is not itself normally distributed, here is a common guideline: For samples of size n greater than 30, the distribution of the sample means can be approximated pretty well by a normal distribution. The approximation gets better as the sample size n becomes larger. • If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size n, not just those with n being greater than 30.

  7. Notation for the Sampling Distribution of • If all possible random samples of size n are selected from a population with mean and standard deviation the mean of the sample means is denoted by , so • Also, the standard deviation of the sample means is denoted by , so

  8. Notes: • When working with an individual value from a normally distributed population, use 2. When working with a mean for some sample, be sure to use the value of for the standard deviation of the sample means. Use

  9. Example • Weights of newborn babies in the United States are normally distributed with a mean of 3420 g and a standard deviation of 495 g. a. If one newborn baby is selected at random, what is the probability that his/her weight is greater than 3600g?

  10. b. If 16 newborn babies are randomly selected, what is the probability that their mean weight is greater than 3700g?

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