t

n = 10 and assume that the sample mean is normally distributed, the chances for estimat- ing incorrect probabilities are great. These mistakes in probability estimates are particu- larly large for sample means in the upper tail of the distribution. Note that the histogram is different from one that would be obtained from a normal distribution. But if you use ampling and Sampling Distributions 39 a random sample of size n = 25, your results are much better. Note that the second his- togram with 1 = 25 is much closer to a normal distribution. The results are even better when the sample size is 50. Thus, even when the distribution of individual observations is highly skewed, the sampling distribution of sample means closely approximates a normal distribution when n 50. The mean and standard deviation for the skewed distribution are 3.3 and 4.247. Thus, the interval from the normal distribution for 95% of the sample means of size 1 = 50 is as follows: 4.247 3.3 (1.96) V50 3.3 $ 1.18 2.12- 4.48 The distribution of sample means for n = 50 appears to fit this interval. From the random sampling studies in this chapter and our previous study of the bino- mial distribution, we have additional evidence to demonstrate the central limit theorem. Similar demonstrations have been produced numerous times by many statisticians. As a result, a large body of empirical evidence supports the application of the central limit theorem to realistic statistical applications, in addition to theoretical results. In Chapter 5 we learned that the binomial random variable has an approximate normal distribution as the sample size becomes large. The question for applied analysis concerns the sample size required to ensure that sample means have a normal distribution. Based on considerable research and experience, we know that, if the distributions are symmetric, then the means from samples of n = 20 to 25 are well approximated by the normal distribution. For skewed distributions the re- quired sample sizes are generally somewhat larger. But note that in the previous examples using a skewed distribution a sample size of n = 50 produced a sampling distribution of sample means that closely followed a normal distribution In this chapter we have begun our discussion of the important statistical problem of making inferences about a population based on results from a sample. The sample mean or sample proportion is often computed to make inferences about population means or proportions. By using the central limit theorem, we have a rationale for applying the tech- niques we develop in future chapters to a wide range of problems. The following exam- ples show important applications of the central limit theorem. Example 6.4 Marketing Study for Antelope Coffee (Normal Probability) Antelope Coffee, Inc., is considering the possibility of opening a gourmet coffee shop in Big Rock, Montana. Previous research has indicated that its shops will be successful in cities of this size if the mean annual family income is above $70,000. It is also assumed that the standard deviation of income is $5,000 in Big Rock, Montana. A random sample of 36 people was obtained, and the mean income was $72,300. Does this sample provide evidence to conclude that a shop should be opened