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Sampling Distributions The sampling distribution of a sample function, say x̄, calculated from a random sample of size n is simply the probability distribution of x̄ (obtained from all possible samples of n observations from a population with mean µ and variance σ 2 ). Central Limit Theorem. If n is large, the sampling distribution of x̄ will be approximately a normal distrubution with mean µx̄ = µ and standard deviation σx̄ = √σn . The standard deviation σx̄ is often referred to as the standard error of the sample mean x̄. Example 1. According to AAA, the average daily meal and lodging costs for a family of four is $213. Assume that the standard deviation of such cost is $15. Consider a random sample of 36 families of four and their travel expenses. Find the ptobability that sample mean exceeds $200. (Ans. 1.0) Example 2. A random sample of size n is to be drawn from a population with µ = 1100 and σ = 200. What size sample would be necessary in order to reduce the standard error of sample mean to 20? (Ans. n = 100) Example 3. The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of .2 ounces. Suppose 100 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 100 bags exceeded 10.45 ounces. (Ans. 0.9938) 4. True or False: The Central Limit Theorem guarantees that the population is normal whenever n is sufficiently large. (False) 5. True or False: The Central Limit Theorem guarantees an approximately normal sampling distribution for the sample mean for large sample sizes, so no knowledge about the distribution of the population is necessary. (True) Other Examples: Exercise 6.34. (Ans. (a) 106; (b) 2.73; (c) Approximately normal) Exercise 6.36. (Ans. 0.7486; 0.2486; (c) 0.7486) Exercise 6.58. (Ans. (a) 98,500.00; (b) 4,242.64; (c) approximately normal; (d) -2.12; (e) 0.983) Exercise 6.59. Example 6. The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 6200 miles. The manufacturer guarantees the tread life of the tires for the first 52,560 miles. (i) What proportion of the tires will need to be replaced under warranty? Ans. 0.1151 (ii) If you buy 36 tires, what is probability that average life of your 36 tires will exceed 58,000? Ans. 0.9738 (iii) The manufacturer is willing to replace only 2% of its tires under a warranty program involving tread life. Find the tread life covered under the warranty. Ans. x0 = 47290 1