6-4 The Central Limit Theorem 285 Central Limit Theorem and the Sampling Distribution of x Given 1. Population (with any distribution) has mean m and standard deviation s. 2. Simple random samples all of the same size n are selected from the population. Practical Rules for Real Applications Involving a Sample Mean x Requirements: Population has a normal distribution or n + 30: Mean of all values of x: mx = m Standard deviation of all values of x: sx = s1 n z score conversion of x: z = x - m s1 n Original population is not normally distributed and n " 30: The distribution of x cannot be approximated well by a normal distribution, and the methods of this section do not apply. Use other methods, such as nonparametric methods (Chapter 13) or bootstrapping methods (Section 7-4). Considerations for Practical Problem Solving 1. Check Requirements: When working with the mean from a sample, verify that the normal distribution can be used by confirming that the original population has a normal distribution or the sample size is n 7 30. 2. Individual Value or Mean from a Sample? Determine whether you are using a normal distribution with a single value x or the mean x from a sample of n values. See the following. • Individual value: When working with an individual value from a normally distributed population, use the methods of Section 6-2 withz = x - m s . • Mean from a sample of values: When working with a mean for some sample of n values, be sure to use the value of s>1n for the standard deviation of the sample means, so use z = x - m s1 n . KEY ELEMENTS The following new notation is used for the mean and standard deviation of the distribution of x. NOTATION FOR THE SAMPLING DISTRIBUTION OF x If all possible simple random samples of size n are selected from a population with mean m and standard deviation s, the mean of all sample means is denoted by mx and the standard deviation of all sample means is denoted by sx. Mean of all values of x: mx = m Standard deviation of all values of x: sx = s2 n Note: sx is called the standard error of the mean and is sometimes denoted as SEM.
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