Section 7.3 Sampling Distributions and the Central Limit Theorem 313 in Figure 7.20, the sample mean is approximately normally distributed even for a small sample size like n = 3. In fact, when a population is normal, the sample mean will also be normal. For a normal population, the sample mean will be normal for any sample size. For a skewed population, the sample size must be large for the sample mean to be approximately normal. Computing the sampling distribution of x for a skewed population For a certain make of car, the number of repairs needed while under warranty has the following probability distribution. x P(x) 0 0.60 1 0.25 2 0.10 3 0.03 4 0.02 Figure 7.22 presents the probability histogram for this distribution, along with probability histograms for the sampling distribution of ̄x for samples of size 3, 10, and 30. The probability histograms for the sampling distributions were created by programming a computer to compute the probability for every possible value of ̄x. 0 1 2 3 4 0.6 0.4 0.2 0 Probability Population 0 1 2 3 4 0.15 0.10 0.05 0 Probability Sample Size n = 3 0.0 0.4 0.8 1.2 1.6 2.0 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 Probability Sample Size n = 10 0 0.25 0.50 0.75 1.00 1.25 0.08 0.06 0.04 0.02 0 Probability Sample Size n = 30 Figure 7.22 The probability histogram for the population distribution is highly skewed. As the sample size increases, the skewness decreases. For a sample size of 30, the probability histogram is reasonably well approximated by a normal curve. The remarkable fact that the sampling distribution of ̄x is approximately normal for a large sample from any distribution is called the Central Limit Theorem. The size of the sample needed to obtain approximate normality depends mostly on the skewness of the population. A sample of size n > 30 is large enough for most populations encountered in practice. Smaller sample sizes are adequate for distributions that are nearly symmetric. The Central Limit Theorem Let ̄x be the mean of a large (n > 30) simple random sample from a population with mean �� and standard deviation ��. Then ̄x has an approximately normal distribution, with mean ��̄ x = �� and standard deviation ��̄ x = �� √ n .
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